ESPN says Real are aware of United's stance on De Gea but will make a bid for him anyway after Gianluigi Donnarumma signed a new contract at AC Milan.
Real also retain an interest in Chelsea goalkeeper Thibaut Courtois, but the Blues have refused to enter into negotiations over the Belgian.
First-choice goalkeeper Keylor Navas's future at Madrid has come under doubt after a string of inconsistent performances last season. However, the Costa Rica international ended the campaign in fine form as he helped Real win a La Liga and Champions League double.
De Gea has two years left on his contract and has not openly agitated for a move away from Old Trafford, strengthening United's hand in any potential negotiations with Real.
United and Real have both set up camp in Los Angeles for pre-season training. The two teams are due to face each other in a friendly in Santa Clara, California on 23 July.
Real came close to signing De Gea from United in the summer of 2015, with Navas moving in the opposite direction as part of the deal. However, the move broke down in the eleventh hour on transfer deadline day.
Navas shrugged off concerns over his Real future in May, saying he was intent on playing for the 12-time European champions for many more seasons.
"I want to stay at Real Madrid for many years to come," he was reported as saying by the Daily Mail.
"I don't care what people say, I just do my job. I notice the fans' support, it makes me feel well-liked."
Real's only signing in the summer transfer window thus far has been French left-back Theo Hernandez from city rivals Atletico Madrid for £25.5m ($33.4m).