The 23-year-old has scored 12 goals in all competitions this season but last started a game on Feb. 20.
Thomas Tuchel replaced Frank Lampard as head coach less than a month earlier and although Abraham has struggled with an ankle problem, he has often failed to make the bench even when fit under the new coach.
ESPN reported in March that contract talks between Chelsea and Abraham had stalled with the England international mindful both of his lack of first-team opportunities and also the club's plans to spend big on a centre-forward this summer.
Abraham will have two years left on his contract at the end of the campaign and sources have told ESPN that Chelsea are willing to offload the striker while they can maximise his value.
Reports have suggested they are prepared to accept £40 million but there is confidence among the interested parties that a deal could be done for less.
West Ham are in the market for a forward while, a return to Villa Park may appeal to Abraham, given he scored 26 goals in a Championship promotion-winning season there in 2018-19 before Lampard gave him his chance at Stamford Bridge.
Abraham is desperate to get his career back on track having fallen out of favour to the extent he is almost certain to miss out on a place in Gareth Southgate's England squad for Euro 2020.
Chelsea are expected to persevere with Timo Werner despite a difficult first season in England but will enter the running for Borussia Dortmund forward Erling Haaland, and are also monitoring Sergio Aguero's departure on a free transfer from Manchester City.