Tierney overruled his assistant Adrian Holmes, who signalled for offside against the City forward.
The Sun says Tierney reviewed the incident on Saturday night and held his hands up that he got it wrong.
The opener by hat-trick hero Sterling allowed City to coast to a 3-1 home win at the Etihad.
The Professional Game Match Officials Limited issued a statement yesterday to state the goal should have been ruled out.
Former Premier League referee Chris Foy delivered his verdict, saying: “It's a difficult decision for the match officials and they've got lots of different considerations.
“But the final outcome should have been an indirect free-kick for offside and the goal shouldn't have stood.
“If he doesn't challenge and the ball just strikes him then the goal would have been allowed, but in this case he's clearly attempting to play the ball so the goal should have been ruled out."