The Ivorian has been tracked by a number of clubs this summer, notably Liverpool, Arsenal and PSG.
But it looks as though United are now in pole position to sign Pepe, who is destined to leave Lille in the coming weeks.
The 23-year-old is currently recovering following the African Cup of Nations.
But as he takes a much-needed summer break, United have stepped up their efforts to land the attacker.
The Times report the Red Devils have progressed in their discussions with Lille over a potential £70m move.
Liverpool and Arsenal have been tracking the player all summer.
But neither side has launched an official bid for Pepe as of yet.
Ole Gunnar Solskjaer is keen on the left-sided attacker and will use the funds from Romelu Lukaku’s Inter move to pay for the transfer.
The Belgian continues to be linked with a move to the Serie A outfit.
But Inter continue to lowball United with their offers for Lukaku.
Ed Woodward wants to recoup around £80m for the striker, which is out of the Italian club’s budget at present.
As for Pepe, his agent Bruno Satin this week refused to rule out a possible move to Napoli.
But he did say a switch to PSG was not on the cards while Neymar is at the club.
“Pepe at Napoli is not impossible, but they have to convince the boy first”, Satin said on Italian radio station Radio Marte’s Si gonfia la rete programme.
“They have everything else demanded by him, a good coach, the Champions League and a good team.
“Pepe deserves a high estimated price, but 90 million euros are too many.
“I believe that Lille will let the boy choose [his next destination] and then find an agreement with the club.
“At PSG there is currently no room for Pepe because there are no clubs that can buy Neymar.”