The 34-year-old moved on a free transfer from PSG on transfer deadline day in October, signing a one-year deal with the option of a further year.
But if the Red Devils opt not to enact that option the Uruguayan - who has scored seven goals for United in all competitions this season - would be due £1.75m (€2m) in compensation, says The Times.
Ole Gunnar Solskjaer and executive vice-chairman Ed Woodward will have to decide whether to extend Cavani's deal, with the United manager said to be keen on keeping the Uruguay international for the 2021-22 season.
While on the pitch he has popped up with important goals - such as a match-winning double at Southampton in November - he has also been a leader off the pitch.
Cavani has scored seven goals in 24 appearances in all competitions for United, the last of which came against Everton on February 6.