The Hammers are looking for attacking reinforcements after seeing Andy Carroll, Lucas Perez, Marko Arnautovic, and Jordan Hugill all depart this summer.
Adebayor is now a free agent after his contract with Turkish side Istanbul Basaksehir was terminated.
And, according to The Sun, the 25-year-old wants a return to England and specifically to London.
Celtic, Watford and Premier League new boys Sheffield United are all said to be considering a move for Adebayor, but West Ham’s location could put them as favourites.
Manuel Pellegrini has already signed striker Sebastian Haller for a club record £45m but is still short in the striker department.
Mexico international Javier Hernandez is the club’s only recognised forward remaining, along with Haller.
While Adebayor is not getting any younger, he knows the Premier League well and is only three goals shy (97) of bagging 100 in England’s top division.
One issue that could come into Pellegrini’s thinking is the fact that Adebayor has not played a full 90 minutes since January.
But he was involved in an ugly scuffle with Galatasaray reserve goalkeeper Ismail Cipe in May.
As the opposition sealed their Turkish Super Lig trophy in a fierce Istanbul derby, Adebayor was shoved to the bench by the shot-stopper.
He managed only three goals in 18 appearances last season but hit 17 in an impressive 2017-18 campaign with the Turkish outfit.
And the forward has a history of controversy from his time in England.
He is probably bets remembered for running the whole length of the pitch to celebrate scoring a goal for Manchester City in front of the Arsenal fans at the Etihad.
Adebayor ha only just joined City from the Gunners that summer, and the away fans reacted angrily to the celebrations.
Signing Adebayor would certainly be a gamble for West Ham but with time running out in the summer transfer window, he might prove to be a useful addition under the circumstances.