The Manchester City goalkeeper is currently on a season-long loan at Torino after being cast aside by Pep Guardiola upon his arrival at the Etihad.
Hart is thought to have a number of Premier League clubs showing interest as his future at City is likely to be over with Claudio Bravo taking up the No1 spot.
And according to The Sun, Slaven Bilic has made Hart the Hammers' priority target for the summer window.
Adrian has failed to live up to expectations this season, and Darren Randolph is yet to convince Bilic he can be their No1 for the foreseeable future.
Joe Hart is attracting interest from West Ham and Chelsea
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The report claims West Ham tried to offload one of them last month to enable them to make a move for Hart.
But with that plan falling flat, Bilic will now wait until the summer to lure the England international to the London Stadium.
A West Ham source said: “We like Hart and would love to bring him to the club.
“That is now one that will have to wait until the summer window. We had more pressing needs in the January window.
Slaven Bilic will make a £15m move for Joe Hart in the summer, according to reports
Adrian is expected to be offloaded at the end of the season
"Darren Randolph has shown form enough that his job is safe until at least the summer when changes will be made.
“The club want to make sure they have enough money in the bank to pay Hart the weekly wages he wants.”
West Ham face competition from Chelsea for Hart's signature however.
Reports suggest Chelsea officials held talks with Hart last month over a prospective switch.
The Hammers are keeping tabs on Asmir Begovic if a move for Hart fails.