United manager Sir Alex Ferguson and chief executive David Gill travelled to a Premier League arbitration tribunal on Monday to argue that the Argentinian should not be allowed to join Liverpool.
The defender has asked for a three-man panel to decide whether United gave him permission to leave the club for an agreed fee.
Heinze wants to join Liverpool and claims he has a letter from United allowing him to leave if a club offered a fee of £6.8million.
The club maintain however that they made it clear to Heinze both orally and by letter that he would not be permitted to join one of their title rivals in the top flight.
A Premier League spokesman confirmed a verdict is likely on Tuesday.