James Rodriguez was left out of Real Madrid's squad for the Champions League final ahead of a summer transfer window in which he is expected to leave the club.
Sky sources understand Manchester United have been offered the chance to buy the attacking-midfielder, though it is understood Real will seek a sum of around £50m for his services.
Speaking last month, Sky Sports' Spanish football expert Guillem Balague also said United appeared to be frontrunners to land the 25-year-old, whose previous start came against Sevilla on May 14.
Rodriguez, who joined Real from Monaco for £71m after helping Colombia reach the quarter-finals of the 2014 World Cup, has struggled for playing time under Zinedine Zidane this season.
And the player, who is contracted at the La Liga champions until 2020, shares the same agent as United boss Jose Mourinho - Jorge Mendes.
Balague said in May: "James is wanted by many teams in Europe and, for me, Manchester United are the favourites to get him. His agent is Jorge Mendes, who is very close to Jose Mourinho.
"He is the kind of player who, instead of going to a club that is guaranteed to win the title next season, would be willing to help a team on the up.
"And Manchester United, with the investment coming in the summer, should be contenders for everything."
United are understood to have cooled their interest in Atletico Madrid's Antoine Griezmann, but are interested in another Real Madrid player, defender Raphael Varane.