The Uruguay international is a free agent after leaving Paris Saint-Germain at the end of last season.
He has told United he still has unfinished business at the age of of 33, despite a glittering career, reports the London Evening Standard.
Ole Gunnar Solskjaer has been desperate to strengthen his squad in what has been a frustrating window.
A one-year deal with the option of a further 12 months is being discussed.
Cavani has been offered to United on more than occasion in the past, but they have previously opted against a move.