Shaw is one of a host of stars on his wanted list - and former United defender May insists the 18-year-old would be ideal.
"There is a different player linked to United every day and that is always going to be the case," he told the Manchester Evening News. "But I think Luke Shaw is a fantastic left-back - the type of player you need. You could have him for the next 10 years easily.
"I think the manager has got to make sure that whoever he brings in is the right type of player.
"You can bring in good players, but you have to play a certain way when you come to United."