The Uruguay forward has been without a club since leaving Paris Saint-Germain at the end of June.
The 33-year-old is PSG's all-time top scorer, with 200 goals in seven seasons at the Ligue 1 champion.
The Daily Mail reported Cavani will sign a two-year deal at Old Trafford worth £210,000 ($380,000) a week after tax.
As a free agent, Cavani is still free to sign for a new club outside the transfer window, but French media reported the deal will be concluded before Tuesday's (AEST) transfer deadline.
Cavani could even make his debut against his former club as United opens its UEFA Champions League campaign away to PSG on 21 October (AEST).
The English giant has been criticised by fans for its lack of activity in the transfer market.
Dutch midfielder Donny van de Beek is United's only signing of the transfer window so far with the Red Devils refusing to match Borussia Dortmund's asking price for long-term target Jadon Sancho.