Ole Gunnar Solskjaer’s position at Old Trafford is coming under pressure after suffering a 1-0 defeat to Arsenal at home.
It means the Red Devils have made their worst start to a Premier League campaign since David Moyes’ reign in the 2013/14 season as they sit 15th.
And Ince, who won two Premier League titles at Old Trafford, believes they will have a big job on their hands if they take over from Solskjaer.
Speaking exclusively to Ladbrokes at its launch of the 5-A-Side Bet on Leicester v Wolves, he said: “Of course any manager would take the Manchester United job. You can't turn that job down.
“Whether it's Pochettino, Max Allegri, if Man United come calling, you're taking the job. There's no two ways about it. Ole's in the hot seat and he's got to make sure he stays there.
“Lots of things need to change at Man United. They need to look at the structure of the club, recruitment, they need a Director of Football.
“Things need to change. A new manager knows that and maybe that's something Pochettino would look at if he was to come in.
“But at the moment Ole's in the hot seat and I hope he stays there. If performances don't improve and results don't come, then all that means is Ole's going to hear names like Pochettino and Allegri a lot more.
“Hopefully it's not the case and I want Ole and Manchester United to do well.
“But when you look at the performance on Sunday, it hardly fills you with confidence, and it makes the next game so, so important. And if they lose that next game, again, those two names are going to crop up.”
The former United player also sent a warning to Solskjaer.
“When the criticism starts to come, that's when other managers' names are being talked about,” Ince added.
“He's heard it all before with Pochettino last season, so he knows he's got to keep on winning. And he did that last year, so credit to him for that.
“But now, again, when results are dropping, then Pochettino's name is going to come up again. Max Allegri.
“You know, when you've got all of these big names out there and you're not getting results, you know they're going to be linked with your job. You can't get away from that.”
But Ince also suggested that the United board “panicked” when they made Solskjaer manager in March 2019.