De Gea competed with Dean Henderson for the No. 1 spot last season and with manager Ole Gunnar Solskjaer yet to make a decision on who will be his first-choice goalkeeper, the Spain international is keen to stake his claim for the gloves rather than immediately look for a move away.
Sources have told ESPN United have yet to receive any concrete interest in the 30-year-old, who still has two years and a 12-month option on his £375,000-a-week contract.
De Gea is set to return late to preseason. He will be granted a holiday after being part of Spain's squad at Euro 2020. United kick off their Premier League campaign against Leeds United at Old Trafford on Aug. 14.
Henderson is set to play a full part in preseason after recovering from the hip injury which forced him out of the England squad.
Solskjaer has four senior goalkeepers in his squad. As well as De Gea and Henderson, third-choice Lee Grant, 38, has signed a new 12-month contract while Tom Heaton, 35, has arrived on a free transfer from Aston Villa.
Heaton has signed a two-year contract with the option of a third.
Sources have told ESPN that United also made an approach for former academy graduate Ben Amos before he joined Wigan Athletic from Charlton Athletic. Amos, 31, was offered the chance to return to Old Trafford, where he graduated from the academy, as No. 3 but chose Wigan instead in order to play regular first team football.
The pair will be part of the squad in Tokyo and could miss the whole of preseason. The men's gold medal match is scheduled for Aug. 7, a week before the start of the new Premier League season.