United chief executive Ed Woodward reportedly contacted his Bayern counterpart Karl-Heinz Ruminigge with a final offer worth £75m.
It comes as Bild report that Liverpool are interested in Muller.
Jurgen Klopp is keeping a keen eye on his situation in Munich as he enters fresh contract negotiations with the German giants.
During the last window Muller admitted that the wages on offer in the Premier League were hard to ignore.
He said: “In the end, we should not forget that playing football is our job.
“So people should accept that wages will always play a role in a player’s decision-making.
“You have to look at the complete package to determine whether something is right for you.
“What’s good for you one day will not necessarily still be good for you the day after.
“I know that a lot of German clubs are unhappy with the Premier League clubs’ spending, but I think it is something good for all clubs in the end.
“It can only be a good thing for football when clubs invest so much money.”