Everton are keen to add competition for Jordan Pickford while Leeds are putting together their transfer plans as they prepare for their first season in the Premier League since 2004 after winning promotion from the Championship.
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Both clubs are interested in Romero, who is No.2 behind David de Gea at Old Trafford. Romero, 33, has an incredible record of 38 clean sheets in 59 games since arriving in 2015 after spells with Racing Club, AZ Alkmaar, Sampdoria and Monaco.
He was unexpectedly left out of Ole Gunnar Solskjaer's team for the FA Cup semifinal against Chelsea, having played in every other round this season.
De Gea was picked in his place and made two crucial mistakes as United slumped to a 3-1 defeat. Romero, capped 96 times by Argentina, is under contract until 2021 and the club hold an option to extend the deal for an additional 12 months.
Meanwhile, sources have confirmed to ESPN that United are looking to make a senior appointment this summer to oversee the club's off-field structure.
Executive vice-chairman Ed Woodward scrapped the idea of appointing a director of football a year ago but the role is back on the agenda, although the new appointment will not just be in charge of overseeing transfers.