Cavani, the Uruguayan striker, will travel to Manchester on Sunday for a medical with a view to announcing the deal by Monday.
The 33-year-old left Paris Saint-Germain this summer after seven years in the French capital and he will now bolster United’s attacking options.
It was reported that talks had ‘accelerated’ between the parties earlier on Saturday but, according to ESPN, everything is now agreed.
Cavani has been widely considered to be one of the world’s top strikers over the past decade after joining PSG from Napoli.
United have been using Anthony Martial in the No. 9 role – a position Marcus Rashford can also operate in – with Odion Ighalo as the primary backup option.
However, manager Ole Gunnar Solskjaer has been keen to add another recognised striker to his squad.
He is also hoping to add a right winger, with Borussia Dortmund star Jadon Sancho his top target.
Ousmane Dembele, the Barcelona attacker, is being considered as an alternative to Sancho.
There could also be outgoings.
Roma have lodged a fresh bid for centre-half Chris Smalling, while Sergio Romero has been linked with an exit after Dean Henderson returned from a loan spell with Sheffield United to challenge David de Gea for the No. 1 spot.