The Spain international played second fiddle to Dean Henderson during the back end of the Premier League campaign.
However, he still got the nod for the Europa League final against Villarreal which United lost through a penalty shoot-out.
The club recently re-signed Tom Heaton for free, and The Times claims that either De Gea or Henderson could be sold.
De Gea has two years remaining on his £375,000-a-week contract, and it is reported that he is not looking for a way out.
The 30-year-old is determined to fight for his place despite struggling for regular game time in the final months of last season.
Paris Saint-Germain were previously fancied to sign him, but they are now on the verge of landing Gianluigi Donnarumma.
The Italy international will join them on a free transfer after running down his contract with Serie A giants AC Milan.