Cavani, a free agent after leaving Paris Saint-Germain at the end of last season, has been offered to every major club in Europe but United are now actively working on an agreement that could see the striker travel to England on Sunday for a medical.
Sources have told ESPN that talks accelerated quickly on Saturday with final discussions scheduled for Sunday morning.
The 33-year-old is keen to move to Old Trafford and could sign an initial one-year deal with the option of another 12 months.
Cavani has already held talks with a number of clubs this summer. The first offer arrived from Benfica, although they weren't able to pay as much as he would have liked.
Juve were the closest to agreeing terms but a disagreement over how long Cavani's contract would be saw the deal break down.
Sources have told ESPN that United do not have concerns about Cavani's age and hope he will be able to have a similar impact to Zlatan Ibrahimovic, who scored 17 Premier League goals in 33 games after joining in 2016, and will provide extra cover once Odion Ighalo's loan ends in January.
United have still not given up hope of landing top target Jadon Sancho before Monday's transfer deadline but time is running out. The club are also working on alternatives including Barcelona's Ousmane Dembele and Sevilla's Lucas Ocampos to ensure manager Ole Gunnar Solskjaer gets the right-winger he wants before the window shuts.
United remain in negotiations with Porto for left-back Alex Telles but have been told by the Portuguese side to up their offer of around €10 million. Porto want nearly double that but United are reluctant to over pay for a player who will be a free agent next summer.
Sources have told ESPN that defenders Chris Smalling and Diogo Dalot could both leave before the deadline. Roma are still hopeful of signing Smalling on a permanent deal after he impressed on loan last season, while AC Milan are interested in a deal for Dalot.
Meanwhile, Lazio have confirmed they have signed midfielder Andreas Pereira on loan. The Italian side will pay all of Pereira's wages for the coming season and have an option to make the deal permanent next summer.