Real Madrid are known to hold a long-standing interest in De Gea, who has made 277 appearances in all competitions for United since leaving Real's city rivals Atletico Madrid in 2011.
It has previously been considered inevitable that the 26-year-old will eventually make the move to the Bernabeu, but according to The Independent, United are confident of tying down the Spaniard to a multiple-year contract.
De Gea has less than two years remaining on his current terms, and the player's value and the club's own position in negotiations will only decrease should no new deal be agreed in the coming months.
However, it has been suggested that United hold the opinion that De Gea is happy with life in Manchester at a time when the club have emerged as strong contenders for the Premier League title.
United sit in second place in the standings, with De Gea conceding just two goals in the first half-a-dozen fixtures of the season.