The 24-year-old has a mammoth £85m buyout clause in his contract but United are hopeful of negotiating the price to a more manageable level.
Manchester United have reportedly sounded out a move for Atletico Madrid keeper Jan Oblak as they fear David de Gea will quit for Real Madrid.
But Romero has already solved a problem about De Gea's successor should all attempts to keep the Spaniard fail.
But despite having potentially played his last game for United, Mourinho has declared he wants to keep De Gea at the club beyond this summer.
He said: "I played one year with Gabbiadini in Sampdoria in Italy and a lot of training".More news: Ducks' John Gibson great but Nashville is for real
More news: India Poll Panel Defends Voting Machines After 'Hacking' Demonstration By Party
More news: Golden State Wins Game 2; Headed To San Antonio
Argentine Romero joined United on a free transfer in 2015 and is set to start next week's Europa League final against Ajax.
Mourinho said that "we have the Argentina national goalkeeper and Spanish national goalkeeper", insisting that he feels that both 'keepers are "phenomenal" and that that the club are in "safe gloves" going forward whatever happens.
"I still think he's the best in the Premier League".
However, it appears as though he isn't the only goalkeeper being linked with a move to Man Utd this summer, as the Manchester Evening News claim that Benfica's Ederson is also on the transfer shortlist.
United manager Jose Mourinho insisted after Wednesday's 0-0 draw at Southampton that De Gea - who did not travel to St Mary's - will still be the club's goalkeeper when they begin their preseason preparations against LA Galaxy in California on July 15. Joel will play against Crystal Palace and Sergio will play in the final.