Summerville, 24, scored five Premier League goals for the Hammers last season and impressed for Holland at the World Cup.
The Dutch were eliminated by Morocco on penalties in the last 32, but Summerville boosted his stock with four goal contributions, two goals and two assists, in as many games at the tournament.
According to the Daily Mail, United bosses have held exploratory talks with their West Ham counterparts to gauge what it would take to sign the winger this summer.
However, a move for Summerville is likely to hinge on Marcus Rashford being sold first, with the England forward seen as a stumbling block to any deal going through.
West Ham United's relegation from the Premier League has weakened their bargaining position, though they are still expected to demand a substantial fee for their left winger.
The Hammers raised £85million by selling Mateus Fernandes to Tottenham, who gazumped United with a higher offer for the midfielder.
Summerville, who previously scored 25 goals in 89 games for Leeds, Manchester United's bitter rivals, is unlikely to come cheap either.
The winger joined West Ham from Leeds in August 2024 for a fee in excess of £25million, and his stock has risen considerably since a difficult first season at the London Stadium that was disrupted by a hamstring injury.
Missing out on Fernandes means United remain in the market for a central midfielder, which continues to be their top priority this summer.
A move for Summerville would add further attacking depth to United's squad, though it remains unclear how quickly a deal could be advanced given the complications around a potential Rashford sale and West Ham's determination to secure a strong fee for one of their most productive attackers from last season.
The pursuit of both a central midfielder and a wide forward means manager Michael Carrick faces a busy few weeks of negotiations before the transfer window closes.