Joe Hart is on loan at Torino
Both the Hammers and Toffees have been heavily linked with the England star who has been shipped out of the Etihad by Pep Guardiola.
But earlier this week Torino's head coach Sinisa Mihajlovic said Hart wanted to stay with the Italian club - on loan - for another season, as reported by Sky Sports.
There is renewed hope for West Ham and Everton this week, though, after Hart dropped an almighty clanger in a 1-1 draw with Crotone at the weekend.
England's Joe Hart makes a spectacular save for England
And speculation in Italy is that the England 'keeper could now be dropped after coming under increasing pressure from fans and media alike.
That could all but seal a Premier League return for Hart.
And it could be a straight choice between the Hammers - who need to find a true number one after nightmare seasons for the departing Adrian and the hapless Darren Randolph - and Toffees for whom neither Maarten Stekelenburg or Joel Robles has shone since Tim Howard's departure.