There was yet another major low for United in their hugely disappointing first season under Moyes on Tuesday night when they were brushed aside 3-0 by derby rivals Manchester City at Old Trafford in the Barclays Premier League.
Ahead of Saturday's top-flight clash with Villa at the same venue, the Red Devils - the reigning champions - are seventh in the table.
And a group of United fans are reportedly trying to raise the money for a plane carrying a banner calling for Moyes' sacking to fly over the ground during the lunchtime contest.
Speaking on Thursday at his pre-match press conference, Lambert branded some of the criticism Moyes has endured as "way over the top", adding that he felt the plane and banner idea was "a bit extreme".
And asked if he thinks Moyes will keep his job, Lambert said: "I would certainly think so.
"I don't think Manchester United as a football club and an organisation will be panicked into doing anything.
"I think the furore is outside the football club. I would fully expect him to be at Manchester United for a while."
Regarding the talk of the plane and banner, Lambert said: "If it was myself, it would not be nice to see.
"He is trying his best for Manchester United, the way I am for Aston Villa.
"It comes with the territory - in the modern game, you are always one bad result away from being absolutely criticised.
"In years gone by you probably would have got a lot of time, but now the success people want is instant.
"But if you are talking about flying an aeroplane over the stadium with a message to say you are not wanted, it is a bit extreme.
"I think some of the criticism is way over the top, I really do.
"I think the magnitude of the job is huge. Only David can truly tell you about that, but from the outside looking in, it is a massive, massive job.
"It is not as if he is standing out there and thinking 'things are not going well, but I still feel good'.
"He is human like everybody else and when you get that level of criticism it would not be nice."
While speaking about Moyes' task in succeeding Sir Alex Ferguson, Lambert made reference to that of the coaches who came in at Borussia Dortmund after Ottmar Hitzfeld.
Lambert, who won the Champions League in 1997 as a Dortmund player under Hitzfeld, said of the situation at United: "Anyone going in there would have found it hard, doesn't matter who you are.
"There have been great managers and you have to try to follow them.
"I remember Dortmund under Ottmar Hitzfeld - it was tough to follow him because of the success he had.
"Even though you don't really want to think about it, it is there at the back of your mind. It happens."
Villa go into Saturday's match lying 12th in the table with eight games left to play, one more than United.
Forward Andreas Weimann looks set to be available for the visitors despite coming off in the first half of Sunday's 4-1 home loss to Stoke due to a dead leg.
But Lambert has reported that Villa midfielder Karim El Ahmadi, who departed the field around the same time, is "struggling" with a thigh problem.
Meanwhile, the manager has given an encouraging update on the progress of striker Libor Kozak in his recovery from the broken leg he suffered in training in January.
"Libor is doing great," Lambert told www.avfc.co.uk.
"He is off the crutches now. I think he's in a much happier place within himself now as well.
"The thing with Libor is he's a top boy so you know that, with his rehabilitation, he'll knuckle down.
"He'll not cut any corners. He'll have good days and bad days because of the mental stress of it but he will do his utmost to get fit.
"He's on course, certainly. We're hoping that by pre-season, he'll not be too far away. Libor is definitely doing well.
"He was down when the whole thing happened and for a few weeks after that.
"But I saw him the other day and we were having a chat. I think coming off the crutches has been a major step considering the amount of time he was on them."