The England midfielder could well prove to be the "attacker" that the Chelsea manager, Jose Mourinho, said he would sign to add to the £8m left-back, Asier del Horno, and the midfielder Steven Gerrard, whose arrival is imminent, to complete his squad for next season.
The Chelsea players report back for pre-season training today and Mourinho is eager to secure his signings as quickly as possible - and certainly before the club's tour to the United States later this month.
Chelsea had also hoped to finalise the future of their striker Hernan Crespo by today. The chief executive, Peter Kenyon, held talks yesterday with Milan, where the Argentinian was on loan last season, but no agreement has been reached. Chelsea want Milan pay a fee for Crespo - who cost them £16.8m two years ago - but the situation has been complicated because Christian Vieri yesterday signed a two-year deal with the Rossoneri after being released by their city rivals, Internazionale.
Ironically, Crespo, who is not due to return to Chelsea until next week because of international commitments, only moved to Stamford Bridge after the club had been let down at the eleventh hour by his then strike partner Vieri. Crespo is desperate to stay in Italy.
It had been thought that Milan were set to sign Parma's Alberto Gilardino - who has also been offered to Chelsea - but the asking price of £19m for the 22-year-old put them off. Chelsea dismissed links to Juventus's David Trezeguet - who has been offered to them as part of a swap involving Crespo. That has also proved unattractive to Chelsea whose pursuit of their first-choice strikers - Andrei Shevchenko, Samuel Eto'o and Adriano - has been fruitless.
If Wright-Phillips signs - and, despite their protestations, City need the cash - it will add to the pressure on Joe Cole as well as provide competition for Arjen Robben and Damien Duff. Mourinho is aware of how dependent he was on the latter two last season and with both susceptible to injury he does not want to go into the new season without options. It would also strengthen the belief that, despite criticism, Didier Drogba would remain as Chelsea's main striker.