Leeds are preparing their first season in the Premier League after a gap of 16 years, and they are expected to bolster their squad in the summer.
Everton, likewise, are looking to sign a solid goalkeeper who can provide competition to Jordan Pickford.
Leeds have signed Illan Meslier on a permanent deal from Lorient. However, Marcelo Bielsa could still be looking to sign an experienced goalkeeper, especially if they allow Kiko Casilla to leave.
Romero, who joined Manchester United on a free transfer in 2015, has kept 38 clean sheets in 59 matches for the Red Devils.
The Argentine goalkeeper has been used mostly as a back-up keeper behind David De Gea. So, he could be tempted to join any Premier League outfit where he can expect regular football.
Romero is on £70k-per-week wages at Manchester United, and that could be a problem for Leeds. However, if they can sort out his wages, he would represent a smart signing for the club.
Everton must aim to sign a back-up goalkeeper this summer. Pickford is very good on his day, but he has lacked consistency.
However, due to the Argentine connection, Leeds, who can offer him regular games, will have a better chance to sign him.