The Cottagers won the Championship at the end of the 2021/22 season to secure automatic promotion to the Premier League.
Yarmolenko, an experienced campaigner, is one of the targets on their transfer wish-list as they prepare for life back in the top flight.
The winger’s current contract with the Hammers will expire this summer and it has emerged that Marco Silva’s side are offering him a one-year deal, with the option of a further year.
The Ukrainian international made 84 appearances for the Hammers in all competitions during his time at the club.
Yarmolenko grabbed the extra-time winner against Europa League specialists Sevilla in the round of 16 as David Moyes’ side put together an exciting run to the semi-finals last season.
Fulham are looking for attacking reinforcements and have managed to hold on to Serbian international Aleksandar Mitrovic, who scored 43 goals for them in the Championship.
The Cottagers are looking to stay in the Premier League for an extended period of time and putting an end to the recent streak of relegation immediately following promotion.