The natural link to Bravo seems unsurprising as the Chilean has had a troubling time in Manchester since arriving from Barcelona in 2016. After being replaced by the impressive Ederson, Bravo has only six appearances for City during the 2017/18 campaign.
However, a deal looks unlikely after Manuel Pellegrini addressed the media back in his home nation, claiming a deal was 'not viable'.
Talking to Encuentro El Mercurio, Pellegrini detailed former transfer struggles at Manchester City when trying to capture the signatures of Alexis Sanchez and Arturo Vidal, also addressing a rumoured move for Bravo.
He said: "It’s true that I tried to sign Alexis Sanchez and Arturo Vidal while at Manchester City, but at that time City was punished by Financial Fair Play and it was impossible.
"I would happily take them to my team, but now for economic reasons, it is impossible. Signing Bravo for West Ham is not viable either."
The Hammers will look to be busy during the transfer window after a season of unrest at the London stadium. Joe Hart recently returned to Manchester City following a difficult loan spell, notably being left out of England's World Cup.
The 31-year-old Hart came in under Slaven Billic last summer but early struggles cost him a place under David Moyes. The former England keeper spent the remaining games of the season on the bench behind Adrian.
West Ham confirmed their first signing of the summer on Tuesday. The London club announced the acquisition of 25-year-old right back Ryan Fredericks on a free transfer from newly promoted Fulham.