The England global is now earning around £130,000-a-week at the Etihad Stadium.
Hart, 30, didn't have the most convincing season on loan in Serie A, combining some energetic performances with an increasing amount of howlers.
West Ham are understood to have struck a deal that will see City pay half of Hart's £100,000-per-week wages, and an option to buy will be included.
West Ham have already signed Argentina defender Pablo Zabaleta from Manchester City.
The move is speeding towards its conclusion and, according to the BBC, Hart will undergo a medical with the Hammers on Monday.
West Ham are on the verge of completing their second signing of the summer with Joe Hart reportedly completing a medical - and the Irons fans on Twitter are excited about the news.
West Ham manager Slaven Bilic has identified bringing in a new first choice goalkeeper as one of his primary targets this summer, leaving Randolph's future at the club in serious doubt.
'He is obviously a fantastic goalie but at the minute he is not our player.
Noble has claimed that Hart "probably deserves a little bit more respect" as the stopper nears a switch to the London Stadium. We have got two really good goalies here so it would be more competition for them.