The 31-year-old made 14 league appearances for the Hammers before his lack of form saw him only start a further five games from December until the end of the campaign.
With rival English goalkeepers producing solid displays in the league this season, according to Sky Sports News, Hart has been informed that he will not be included in the England 23-man squad heading to Russia this summer.
Manchester City boss Pep Guardiola has already made it clear that he sees no future for Hart at the club anymore, meaning that the Englishman will once again need to move to try and regain the form that has led to his 75 England caps.
With a move to a top-six club unlikely, Hart will need to move to a club where his status as the first-choice goalkeeper is assured.
Newly promoted Wolverhampton Wanderers may present an attractive option for Hart. Club boss Nuno Espírito Santo has already spoken about having to make big changes to adapt to life in the Premier League next season.
The club's current number one John Ruddy isn't as experienced as Hart in the Premier League, while Hart himself may find less pressure playing in a side with the ambition of consolidation rather than expectation.
With one year left on his City contract, potential transfer fees and wages would be affordable for Wolves who are backed by wealthy Chinese investors Fosun Group.