diff --git a/pkg/kv/kvserver/txnwait/queue.go b/pkg/kv/kvserver/txnwait/queue.go
index fda0c7fb0528..b09558d3604c 100644
--- a/pkg/kv/kvserver/txnwait/queue.go
+++ b/pkg/kv/kvserver/txnwait/queue.go
@@ -84,8 +84,18 @@ func ShouldPushImmediately(req *roachpb.PushTxnRequest) bool {
 // CanPushWithPriority returns true if the given pusher can push the pushee
 // based on its priority.
 func CanPushWithPriority(pusher, pushee enginepb.TxnPriority) bool {
-	return pusher > pushee &&
-		(pusher == enginepb.MaxTxnPriority || pushee == enginepb.MinTxnPriority)
+	// Normalize the transaction priorities so that normal user priorities are
+	// considered equal for the purposes of pushing.
+	normalize := func(p enginepb.TxnPriority) enginepb.TxnPriority {
+		switch p {
+		case enginepb.MinTxnPriority, enginepb.MaxTxnPriority:
+			return p
+		default /* normal txn priorities */ :
+			return enginepb.MinTxnPriority + 1
+		}
+	}
+	pusher, pushee = normalize(pusher), normalize(pushee)
+	return pusher > pushee
 }
 
 // isPushed returns whether the PushTxn request has already been
diff --git a/pkg/kv/kvserver/txnwait/queue_test.go b/pkg/kv/kvserver/txnwait/queue_test.go
index 131bdef457e2..11fc1fce4877 100644
--- a/pkg/kv/kvserver/txnwait/queue_test.go
+++ b/pkg/kv/kvserver/txnwait/queue_test.go
@@ -12,6 +12,7 @@ package txnwait
 
 import (
 	"context"
+	"fmt"
 	"sync"
 	"sync/atomic"
 	"testing"
@@ -33,7 +34,8 @@ func TestShouldPushImmediately(t *testing.T) {
 
 	min := enginepb.MinTxnPriority
 	max := enginepb.MaxTxnPriority
-	mid := enginepb.TxnPriority(1)
+	mid1 := enginepb.TxnPriority(1)
+	mid2 := enginepb.TxnPriority(2)
 	testCases := []struct {
 		force      bool
 		typ        roachpb.PushTxnType
@@ -42,59 +44,101 @@ func TestShouldPushImmediately(t *testing.T) {
 		shouldPush bool
 	}{
 		{false, roachpb.PUSH_ABORT, min, min, false},
-		{false, roachpb.PUSH_ABORT, min, mid, false},
+		{false, roachpb.PUSH_ABORT, min, mid1, false},
+		{false, roachpb.PUSH_ABORT, min, mid2, false},
 		{false, roachpb.PUSH_ABORT, min, max, false},
-		{false, roachpb.PUSH_ABORT, mid, min, true},
-		{false, roachpb.PUSH_ABORT, mid, mid, false},
-		{false, roachpb.PUSH_ABORT, mid, max, false},
+		{false, roachpb.PUSH_ABORT, mid1, min, true},
+		{false, roachpb.PUSH_ABORT, mid1, mid1, false},
+		{false, roachpb.PUSH_ABORT, mid1, mid2, false},
+		{false, roachpb.PUSH_ABORT, mid1, max, false},
+		{false, roachpb.PUSH_ABORT, mid2, min, true},
+		{false, roachpb.PUSH_ABORT, mid2, mid1, false},
+		{false, roachpb.PUSH_ABORT, mid2, mid2, false},
+		{false, roachpb.PUSH_ABORT, mid2, max, false},
 		{false, roachpb.PUSH_ABORT, max, min, true},
-		{false, roachpb.PUSH_ABORT, max, mid, true},
+		{false, roachpb.PUSH_ABORT, max, mid1, true},
+		{false, roachpb.PUSH_ABORT, max, mid2, true},
 		{false, roachpb.PUSH_ABORT, max, max, false},
 		{false, roachpb.PUSH_TIMESTAMP, min, min, false},
-		{false, roachpb.PUSH_TIMESTAMP, min, mid, false},
+		{false, roachpb.PUSH_TIMESTAMP, min, mid1, false},
+		{false, roachpb.PUSH_TIMESTAMP, min, mid2, false},
 		{false, roachpb.PUSH_TIMESTAMP, min, max, false},
-		{false, roachpb.PUSH_TIMESTAMP, mid, min, true},
-		{false, roachpb.PUSH_TIMESTAMP, mid, mid, false},
-		{false, roachpb.PUSH_TIMESTAMP, mid, max, false},
+		{false, roachpb.PUSH_TIMESTAMP, mid1, min, true},
+		{false, roachpb.PUSH_TIMESTAMP, mid1, mid1, false},
+		{false, roachpb.PUSH_TIMESTAMP, mid1, mid2, false},
+		{false, roachpb.PUSH_TIMESTAMP, mid1, max, false},
+		{false, roachpb.PUSH_TIMESTAMP, mid2, min, true},
+		{false, roachpb.PUSH_TIMESTAMP, mid2, mid1, false},
+		{false, roachpb.PUSH_TIMESTAMP, mid2, mid2, false},
+		{false, roachpb.PUSH_TIMESTAMP, mid2, max, false},
 		{false, roachpb.PUSH_TIMESTAMP, max, min, true},
-		{false, roachpb.PUSH_TIMESTAMP, max, mid, true},
+		{false, roachpb.PUSH_TIMESTAMP, max, mid1, true},
+		{false, roachpb.PUSH_TIMESTAMP, max, mid2, true},
 		{false, roachpb.PUSH_TIMESTAMP, max, max, false},
 		{false, roachpb.PUSH_TOUCH, min, min, true},
-		{false, roachpb.PUSH_TOUCH, min, mid, true},
+		{false, roachpb.PUSH_TOUCH, min, mid1, true},
+		{false, roachpb.PUSH_TOUCH, min, mid2, true},
 		{false, roachpb.PUSH_TOUCH, min, max, true},
-		{false, roachpb.PUSH_TOUCH, mid, min, true},
-		{false, roachpb.PUSH_TOUCH, mid, mid, true},
-		{false, roachpb.PUSH_TOUCH, mid, max, true},
+		{false, roachpb.PUSH_TOUCH, mid1, min, true},
+		{false, roachpb.PUSH_TOUCH, mid1, mid1, true},
+		{false, roachpb.PUSH_TOUCH, mid1, mid2, true},
+		{false, roachpb.PUSH_TOUCH, mid1, max, true},
+		{false, roachpb.PUSH_TOUCH, mid2, min, true},
+		{false, roachpb.PUSH_TOUCH, mid2, mid1, true},
+		{false, roachpb.PUSH_TOUCH, mid2, mid2, true},
+		{false, roachpb.PUSH_TOUCH, mid2, max, true},
 		{false, roachpb.PUSH_TOUCH, max, min, true},
-		{false, roachpb.PUSH_TOUCH, max, mid, true},
+		{false, roachpb.PUSH_TOUCH, max, mid1, true},
+		{false, roachpb.PUSH_TOUCH, max, mid2, true},
 		{false, roachpb.PUSH_TOUCH, max, max, true},
 		// Force pushes always push immediately.
 		{true, roachpb.PUSH_ABORT, min, min, true},
-		{true, roachpb.PUSH_ABORT, min, mid, true},
+		{true, roachpb.PUSH_ABORT, min, mid1, true},
+		{true, roachpb.PUSH_ABORT, min, mid2, true},
 		{true, roachpb.PUSH_ABORT, min, max, true},
-		{true, roachpb.PUSH_ABORT, mid, min, true},
-		{true, roachpb.PUSH_ABORT, mid, mid, true},
-		{true, roachpb.PUSH_ABORT, mid, max, true},
+		{true, roachpb.PUSH_ABORT, mid1, min, true},
+		{true, roachpb.PUSH_ABORT, mid1, mid1, true},
+		{true, roachpb.PUSH_ABORT, mid1, mid2, true},
+		{true, roachpb.PUSH_ABORT, mid1, max, true},
+		{true, roachpb.PUSH_ABORT, mid2, min, true},
+		{true, roachpb.PUSH_ABORT, mid2, mid1, true},
+		{true, roachpb.PUSH_ABORT, mid2, mid2, true},
+		{true, roachpb.PUSH_ABORT, mid2, max, true},
 		{true, roachpb.PUSH_ABORT, max, min, true},
-		{true, roachpb.PUSH_ABORT, max, mid, true},
+		{true, roachpb.PUSH_ABORT, max, mid1, true},
+		{true, roachpb.PUSH_ABORT, max, mid2, true},
 		{true, roachpb.PUSH_ABORT, max, max, true},
 		{true, roachpb.PUSH_TIMESTAMP, min, min, true},
-		{true, roachpb.PUSH_TIMESTAMP, min, mid, true},
+		{true, roachpb.PUSH_TIMESTAMP, min, mid1, true},
+		{true, roachpb.PUSH_TIMESTAMP, min, mid2, true},
 		{true, roachpb.PUSH_TIMESTAMP, min, max, true},
-		{true, roachpb.PUSH_TIMESTAMP, mid, min, true},
-		{true, roachpb.PUSH_TIMESTAMP, mid, mid, true},
-		{true, roachpb.PUSH_TIMESTAMP, mid, max, true},
+		{true, roachpb.PUSH_TIMESTAMP, mid1, min, true},
+		{true, roachpb.PUSH_TIMESTAMP, mid1, mid1, true},
+		{true, roachpb.PUSH_TIMESTAMP, mid1, mid2, true},
+		{true, roachpb.PUSH_TIMESTAMP, mid1, max, true},
+		{true, roachpb.PUSH_TIMESTAMP, mid2, min, true},
+		{true, roachpb.PUSH_TIMESTAMP, mid2, mid1, true},
+		{true, roachpb.PUSH_TIMESTAMP, mid2, mid2, true},
+		{true, roachpb.PUSH_TIMESTAMP, mid2, max, true},
 		{true, roachpb.PUSH_TIMESTAMP, max, min, true},
-		{true, roachpb.PUSH_TIMESTAMP, max, mid, true},
+		{true, roachpb.PUSH_TIMESTAMP, max, mid1, true},
+		{true, roachpb.PUSH_TIMESTAMP, max, mid2, true},
 		{true, roachpb.PUSH_TIMESTAMP, max, max, true},
 		{true, roachpb.PUSH_TOUCH, min, min, true},
-		{true, roachpb.PUSH_TOUCH, min, mid, true},
+		{true, roachpb.PUSH_TOUCH, min, mid1, true},
+		{true, roachpb.PUSH_TOUCH, min, mid2, true},
 		{true, roachpb.PUSH_TOUCH, min, max, true},
-		{true, roachpb.PUSH_TOUCH, mid, min, true},
-		{true, roachpb.PUSH_TOUCH, mid, mid, true},
-		{true, roachpb.PUSH_TOUCH, mid, max, true},
+		{true, roachpb.PUSH_TOUCH, mid1, min, true},
+		{true, roachpb.PUSH_TOUCH, mid1, mid1, true},
+		{true, roachpb.PUSH_TOUCH, mid1, mid2, true},
+		{true, roachpb.PUSH_TOUCH, mid1, max, true},
+		{true, roachpb.PUSH_TOUCH, mid2, min, true},
+		{true, roachpb.PUSH_TOUCH, mid2, mid1, true},
+		{true, roachpb.PUSH_TOUCH, mid2, mid2, true},
+		{true, roachpb.PUSH_TOUCH, mid2, max, true},
 		{true, roachpb.PUSH_TOUCH, max, min, true},
-		{true, roachpb.PUSH_TOUCH, max, mid, true},
+		{true, roachpb.PUSH_TOUCH, max, mid1, true},
+		{true, roachpb.PUSH_TOUCH, max, mid2, true},
 		{true, roachpb.PUSH_TOUCH, max, max, true},
 	}
 	for _, test := range testCases {
@@ -111,9 +155,46 @@ func TestShouldPushImmediately(t *testing.T) {
 					Priority: test.pusheePri,
 				},
 			}
-			if shouldPush := ShouldPushImmediately(&req); shouldPush != test.shouldPush {
-				t.Errorf("expected %t; got %t", test.shouldPush, shouldPush)
-			}
+			shouldPush := ShouldPushImmediately(&req)
+			require.Equal(t, test.shouldPush, shouldPush)
+		})
+	}
+}
+
+func TestCanPushWithPriority(t *testing.T) {
+	defer leaktest.AfterTest(t)()
+
+	min := enginepb.MinTxnPriority
+	max := enginepb.MaxTxnPriority
+	mid1 := enginepb.TxnPriority(1)
+	mid2 := enginepb.TxnPriority(2)
+	testCases := []struct {
+		pusher enginepb.TxnPriority
+		pushee enginepb.TxnPriority
+		exp    bool
+	}{
+		{min, min, false},
+		{min, mid1, false},
+		{min, mid2, false},
+		{min, max, false},
+		{mid1, min, true},
+		{mid1, mid1, false},
+		{mid1, mid2, false},
+		{mid1, max, false},
+		{mid2, min, true},
+		{mid2, mid1, false},
+		{mid2, mid2, false},
+		{mid2, max, false},
+		{max, min, true},
+		{max, mid1, true},
+		{max, mid2, true},
+		{max, max, false},
+	}
+	for _, test := range testCases {
+		name := fmt.Sprintf("pusher=%d/pushee=%d", test.pusher, test.pushee)
+		t.Run(name, func(t *testing.T) {
+			canPush := CanPushWithPriority(test.pusher, test.pushee)
+			require.Equal(t, test.exp, canPush)
 		})
 	}
 }
@@ -158,9 +239,8 @@ func TestIsPushed(t *testing.T) {
 					WriteTimestamp: test.txnTimestamp,
 				},
 			}
-			if isPushed := isPushed(&req, &txn); isPushed != test.isPushed {
-				t.Errorf("expected %t; got %t", test.isPushed, isPushed)
-			}
+			isPushed := isPushed(&req, &txn)
+			require.Equal(t, test.isPushed, isPushed)
 		})
 	}
 }