Both The Sun and the Daily Mirror revealed that the Blues had agreed a five-year deal with the 23-year-old England international.
City boss Stuart Pearce had hoped to keep the highly-rated Wright-Phillips, who signed a new contract last season.
But the club are heavily in debt and could not refuse the offer.
Wright-Phillips joined City in the summer of 1998 having been a trainee at Nottingham Forest and has played almost 150 games for the club.
Arsenal were also thought to be keen to sign him but could not compete with the spending power of Roman Abramovich.
No Chelsea bid yet for Wright-Phillips · Update
Manchester City have yet to receive a bid from Chelsea for winger Shaun Wright-Phillips.
The Blues are reported to have offered City £18million for the 23-year-old England international.
However the Press Association understands that City have heard nothing yet from Stamford Bridge.