Boss Ole Gunnar Solskjaer is yet to name a permanent captain following the departure of former skipper Antonio Valencia at the end of last season.
De Gea, who has spent eight years at Old Trafford, has put himself forward as a candidate for the role.
"I've been captain for some games," he said. "Of course, it is amazing to be captain of a club like Manchester United and defend this badge. It's amazing, so of course I'll be really, really happy to [be] captain."
De Gea has been one of United's standout performers during the difficult years following Alex Ferguson's retirement in 2013, the last time they won the Premier League.
He has been named the club's player of the season on four occasions and has been in the PFA Team of the Year five times.
However, some high-profile mistakes undermined his personal contribution to a difficult season in 2018-19, in which United finished sixth in the league and failed to win a trophy, and De Gea admits there is much work to be done to take the team back to where they believe they belong.
"We have to improve a lot. We are Manchester United; we need to fight for trophies," he said.