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, scoring 200 goals in seven seasons at the Parc des Princes.
The Daily Mail reported Cavani will sign a two-year deal at Old Trafford worth £210,000 ($270,000) 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 Monday's transfer deadline.
Cavani could even make his debut against his former club as United open their Champions League campaign away to PSG on October 20.
The English giants have been criticised by fans for their 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.
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