The 33-year-old Uruguay international had been without a club since June after ending his seven-year stay with the Ligue 1 champions.
United have sought to sign short-term cover for striker Anthony Martial with Odion Ighalo's loan from Shanghai Shenhua set to expire at the end of January.
The Old Trafford club consider Cavani to be a proven winner who fits into the recruitment philosophy of combining high-quality recruits with homegrown talent.
It is hoped that the Uruguayan will be able to support the development of younger players, particularly Martial, Marcus Rashford and Mason Greenwood in United's attack.
Cavani arrives at Old Trafford having scored 341 goals in a 556-game club career, winning six Ligue 1 titles at PSG and the Copa America with Uruguay in 2011.
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