The 23-year-old has become the club's number one following an inconsistent start to his United career after arriving from Atletico Madrid in 2011.
The Spaniard has two years remaining on his contract, but Moyes, who is expected to have more than £100m to spend on players in the summer, has suggested that fresh terms could be agreed.
According to PA, when asked if De Gea's contract could be extended, Moyes said: "The club are always going to keep their best players. The club will always look to keep the players who they feel they need to."
United are currently seventh in the Premier League table, 12 points adrift of a Champions League spot.