The Netherlands winger has attracted interest from top clubs around Europe this summer, including Manchester United, but Roma are looking to move ahead in the race, per Sky Sports.
The Red Devils would only make a move for Summerville should Marcus Rashford depart the club this summer, with the Englishman's future currently up in the air.
It is believed that several other Premier League clubs are interested in the Dutch winger.
Summerville impressed for the Netherland's at this summer's World Cup, scoring four goals while providing two assists in four matches.
He made 31 appearances for The Hammers last season, netting five times and registering two assists.
Following West Ham's relegation to the Championship, Summerville is expected to leave the club in order to either play in Italy or make a return to the Premier League.