The Sun says Amorim could have his non-international players back in training this month.
The Portuguese wants personal bests from each player in terms of fitness by mid-July.
And if they fail to reach standards set, then those players will be sidelined come the start of the season.
Amorim is so determined to turn things around that he has even insisted he know where each player is holidaying this summer break.
A club source said: “Amorim needs a good pre-season. He’s not shied away from the pressure he’s under.
“He would have always preferred to have arrived at United ahead of a new season rather than in the middle of it.
“He has to get this right. He’s told the players there are no more excuses. His job is on the line — and so too is theirs.
“The club can’t accept any more failures.”