The winger is expected to leave West Ham following their relegation from the Premier League, with interest coming from several top-flight clubs.
Chelsea and Manchester United have both been monitoring Summerville, though it remains to be seen whether the Netherlands international earns a move to one of the division's leading sides.
Sources have suggested a more realistic outcome could see the firmest interest come from an ambitious side outside the elite.
Fulham need to strengthen their wide options after losing Harry Wilson, who is poised to join Leeds United on a free transfer, and have added Summerville to their shortlist.
It remains unclear whether Fulham, who are expected to appoint Alvaro Arbeloa as their new manager, will trigger an option to make Samuel Chukwueze's loan from AC Milan permanent.
Summerville impressed for the Netherlands at the World Cup, scoring two goals in four games. He failed to convert the decisive spot-kick, though, as Ronald Koeman's side lost on penalties to Morocco in the last 32.
The 24-year-old was also one of West Ham's best players last season, scoring seven goals in his final ten games to lift the club's hopes of avoiding relegation.
Doubts remain over his consistency and physicality at the highest level.
The former Leeds United winger struggled for fitness after joining West Ham in the summer of 2024, missing much of the following season with a hamstring injury and a chunk of last season with a calf problem.
West Ham United accept they have little chance of keeping Summerville.
The club have already raised funds by selling Mateus Fernandes to Tottenham for £85 million.
West Ham's aim is to leave Nuno Espirito Santo with a squad capable of winning promotion at the first attempt, with the Fernandes and Summerville sales intended to ease the pressure to sell captain Jarrod Bowen, who continues to attract Premier League interest.