The Hammers have joined a host of other clubs who are exploring the possibilities of a move for the England midfielder.
Phillips is also a target for Newcastle United, who have suffered from a host of injuries to midfielders, as well as having Sandro Tonali banned for 10 months for gambling on matches during his time with AC Milan.
Crystal Palace manager Roy Hodgson has admitted he would like to bring Phillips to Selhurst Park as he looks for a replacement for the injured Cheikh Doucoure.
Juventus have also held discussions with City about the 28-year-old, though the end of a tax break which had helped Serie A clubs take foreigners to Italy would make it more expensive for them than it had been to sign the midfielder.
City want a £7m loan fee for Phillips, which has proved a deterrent so far and is an issue for Newcastle, who have limited leeway within Profit and Sustainability Rules.
West Ham were interested in Phillips in the summer, when he instead chose to stay at City and David Moyes bought two other midfielders, in Edson Alvarez and James Ward-Prowse.
Phillips has not started a Premier League game this season and has only begun six matches in all competitions since his £42.5m move to City in 2022.
Pep Guardiola is willing to let the former Leeds player leave on loan to help him bolster his chances of retaining his place in the England squad for Euro 2024, providing City get an agreement that suits them.