Assistant coach Gary Neville will delay his arrival after his Valencia side lost a third La Liga game in a row.
Hart and Sterling could be out for four weeks, making them serious doubts for City's Champions League quarter-final.
Winger Sterling has a groin injury, while keeper Hart has a calf problem.
Sterling, 21, limped off during City's 1-0 loss to United, while Hart, 28, was hurt after clearing a poor backpass from Martin Demichelis and went off on a stretcher.
City boss Manuel Pellegrini, asked whether they would miss England's games in Germany on Saturday and against the Netherlands at Wembley on 29 March, said: "I don't think muscle injuries are one week. Normally this is three or four weeks."
Neville, meanwhile, is to stay in Valencia to work with his players after fans called for the ex-England defender to leave following a 2-0 home defeat by Celta Vigo.
The 41-year-old, who said he will link up with the England squad on Thursday, said: "You'll report it as panic, but it is not panic, it is something that's important.
"I have just spoken to Roy Hodgson and I am going to stay back in Valencia. The players train Tuesday, Wednesday, Thursday this week. I feel it's the right thing to do.
"Obviously it's not an ideal situation. I'm the assistant coach for the national team of England, with 60 million people, but I understand the two or three million people in Valencia need my attention this week."
Stoke City midfielder Ibrahim Afellay and Sunderland defender Patrick van Aanholt could feature against England at Wembley after being added to the Netherlands squad.
Coach Danny Blind has been hampered by the withdrawal of several players for next week's friendly matches against France and England.
PSV Eindhoven pair Davy Propper and Jurgen Locadia pulled out on Monday, missing out along with Arjen Robben, Daryl Janmaat, Erik Pieters and Ron Vlaar.
Afellay was a surprise omission from the original squad, while Van Aanholt could make his first international appearance in more than two years.
The Netherlands were World Cup semi-finalists in 2014, but have failed to qualify for Euro 2016.
Joe Hart and Raheem Sterling are set to miss England's friendlies with Germany and the Netherlands after being injured during Manchester City's derby defeat by Manchester United on Sunday.