diff --git a/http2/transport.go b/http2/transport.go
index 54acc1e36..7a07ad591 100644
--- a/http2/transport.go
+++ b/http2/transport.go
@@ -668,6 +668,7 @@ func (t *Transport) newClientConn(c net.Conn, singleUse bool) (*ClientConn, erro
 	cc.inflow.add(transportDefaultConnFlow + initialWindowSize)
 	cc.bw.Flush()
 	if cc.werr != nil {
+		cc.Close()
 		return nil, cc.werr
 	}
 
@@ -1033,6 +1034,15 @@ func (cc *ClientConn) roundTrip(req *http.Request) (res *http.Response, gotErrAf
 	bodyWriter := cc.t.getBodyWriterState(cs, body)
 	cs.on100 = bodyWriter.on100
 
+	defer func() {
+		cc.wmu.Lock()
+		werr := cc.werr
+		cc.wmu.Unlock()
+		if werr != nil {
+			cc.Close()
+		}
+	}()
+
 	cc.wmu.Lock()
 	endStream := !hasBody && !hasTrailers
 	werr := cc.writeHeaders(cs.ID, endStream, int(cc.maxFrameSize), hdrs)
diff --git a/http2/transport_test.go b/http2/transport_test.go
index 1424f818b..599525418 100644
--- a/http2/transport_test.go
+++ b/http2/transport_test.go
@@ -4568,3 +4568,65 @@ func TestClientConnTooIdle(t *testing.T) {
 		}
 	}
 }
+
+type fakeConnErr struct {
+	net.Conn
+	writeErr error
+	closed   bool
+}
+
+func (fce *fakeConnErr) Write(b []byte) (n int, err error) {
+	return 0, fce.writeErr
+}
+
+func (fce *fakeConnErr) Close() error {
+	fce.closed = true
+	return nil
+}
+
+// issue 39337: close the connection on a failed write
+func TestTransportNewClientConnCloseOnWriteError(t *testing.T) {
+	tr := &Transport{}
+	writeErr := errors.New("write error")
+	fakeConn := &fakeConnErr{writeErr: writeErr}
+	_, err := tr.NewClientConn(fakeConn)
+	if err != writeErr {
+		t.Fatalf("expected %v, got %v", writeErr, err)
+	}
+	if !fakeConn.closed {
+		t.Error("expected closed conn")
+	}
+}
+
+func TestTransportRoundtripCloseOnWriteError(t *testing.T) {
+	req, err := http.NewRequest("GET", "https://dummy.tld/", nil)
+	if err != nil {
+		t.Fatal(err)
+	}
+	st := newServerTester(t, func(w http.ResponseWriter, r *http.Request) {}, optOnlyServer)
+	defer st.Close()
+
+	tr := &Transport{TLSClientConfig: tlsConfigInsecure}
+	defer tr.CloseIdleConnections()
+	cc, err := tr.dialClientConn(st.ts.Listener.Addr().String(), false)
+	if err != nil {
+		t.Fatal(err)
+	}
+
+	writeErr := errors.New("write error")
+	cc.wmu.Lock()
+	cc.werr = writeErr
+	cc.wmu.Unlock()
+
+	_, err = cc.RoundTrip(req)
+	if err != writeErr {
+		t.Fatalf("expected %v, got %v", writeErr, err)
+	}
+
+	cc.mu.Lock()
+	closed := cc.closed
+	cc.mu.Unlock()
+	if !closed {
+		t.Fatal("expected closed")
+	}
+}