It's broke today that United have made an improved offer of £60m plus bonuses to Brentford for the Cameroon forward.
Asked if Mbeumo can handle the pressure of succeeding at Old Trafford, Pallister told Tribalfootball.com: “Well, that's the million-dollar question. That's why they are all sitting there now trying to get players across, and Bryan Mbeumo, that’s why there's talk about him going with Thomas Frank to Tottenham.
"They finished below us last season, so maybe we got the edge on that one. But no, I think he's a good player. He looks like a very calm character, and if he were to come to Old Trafford, I think he would embrace it and could be a real threat.
“It's not always a gimme because for Manchester United, it is a little bit different from playing everywhere else. As I say in what we're seeing now are difficult times. The squad has got a few issues, there are players coming and going, there's a lot of talk about a total rebound of the squad, so it's going to be an interesting few months leading up.
"Whatever happens, I hope they can get the work done early, keep the team together, get the squad together and ready for the pre-season, and everybody gets a chance to know each other.
“But it never works out as easily as that. I'm sure there's going to be a few hiccups along the way. It'd be nice to have a settled idea about what your team's going to be when you go on your pre-season tour.”
Academy shake-up a concern, but not priority
Meanwhile, over the past week Everton have named former United academy director Nick Cox as their new technical director. Cox's decision to leave United comes just weeks after U18 coach Adam Lawrence made the same choice.
Asked if the departures were a sign of major issues behind-the-scenes at Carrington, Pallister concedes the instability is a concern.
He continued: “I really don't know all the workings of the academy, but the academy's been a big part of Manchester United for a long time, and I don't know what's going on behind the scenes, whether they've been poached, whether there's a fallout or anything, I really don't know but nobody likes instability.
"You strike up relationships with families and kids as they’re coming through the school careers and coming to Manchester United, and all of a sudden, that's not going to be there.
“Hopefully, they can find somebody to fit into those roles that are going to be equally as good. I'm guessing in the list of the biggest problems, that's maybe not right up there, but it is I suppose a concern that somebody who's being in charge of academy now left because everybody likes a bit of continuity and everybody likes to get to know the people, your parents like to get to know the people who looking after their kids when they're going to the academy at a young age.
"It's a concern, but as I say, on a list of priorities that we've got at the moment, it's not maybe right up there.”
- Gary Pallister was speaking to Tribalfootball on behalf of Gambleonline.net