The two clubs are believed to be in talks over a potential move for the 32-year-old, as City look to bolster their midfield.
The Eastlands club's boss Roberto Mancini had previously made it known that he did not envisage any new signings on deadline day, but maintained that he remained interested in adding to his midfield.
Now former Barcelona B boss Enrique has confirmed that whilst the deal has yet to be completed, it is likely Pizarro will join the Premier League leaders by the end of the transfer window.
"He hasn't left yet, but when it happens we'll let you know," Enrique told reporters.
If he leaves [Rome], maybe something has happened. I always try to be correct, I always talk to players to their faces.
"My job is to choose players. Pizarro has always behaved well, but I must make these choices."
Meanwhile, the midfielder's agent, Beppe Bozzo, has confirmed that despite initial concerns over his work permit, the player will fly to Manchester this afternoon.
Bozzo said: "The relative problems in the work permit have been overcome and Manchester City will request dispensation relative to his national team appearances. We will fly to England at 2pm [1pm GMT]."It has also been reported that Genoa, Juventus and Zenit St Petersburg are interested in signing the Chilean, who has previously worked with the City boss during his short stint at Inter.
Should the potential transfer go through, the 32-year-old would add a new dimension to City's midfield with his dominating game, but his lack of time on the pitch at Roma may mean the player needs to time to build up his match fitness.
Pizarro has made only seven appearances for Roma this season, and it is believed that both the club and player wish to part ways.