Moyes's 10-month stint in charge of the Red Devils was brought to an end last April following a campaign which saw the champions slip alarmingly off the pace in the Premier League.
Louis van Gaal was appointed as new manager in the summer, although Allardyce says that former Everton boss Moyes should have been supported in the same way as the new man at the helm.
"There was a complacency by United in not going out and delivering the signings David felt he needed," The Sun quotes the 59-year-old as saying. "Now there's a panic on. He will obviously look at what he might have done better but he should have got the players he wanted and he tells me he didn't get any of them.
"It was difficult enough taking over from Alex Ferguson but if, when you take over, you don't get what you want, it's so much harder. It was a great shame because he'd done everything right at Everton and Sir Alex saw the fact he wanted to build at United like he had done at Everton.
"I'm not so sure anybody would have been successful in that season - history tells you that."
United are currently 12th in the league table following a slow start to life under Van Gaal.