And journalist Duncan Castles claims they have held further discussions with Lille about a potential transfer.
The Red Devils first enquired about the pacy, highly-coveted attacker back in May.
While Manchester United have held off on pursuing a move, they are now stepping up their hunt for Pepe.
The Lille star has been linked with a host of top clubs across Europe this summer.
And he does tick the boxes for Ole Gunnar Solskjaer as a young, hungry and speedy talent.
Arsenal and Everton have asked about the 24-year-old, but it is United who have looked into the finances over a deal.
Solskjaer’s team are said to have approached Lille to uncover what wages Pepe would demand and what his asking price would be.
A £70m price-tag has been mooted, and Lille are increasingly optimistic they can lure enough interest in their goalscoring forward to get that fee.
“United I told you back in May had made their first enquiry direct to the club about Pepe”
Speaking on the latest episode of the Transfer Window Podcast, Castles said: “Lille are a lot more confident they’ll achieve the €80m asking price they’ve been looking for for Pepe.
“He’s just finished playing the Africa Cup of Nations, and I think that’s a factor in the delay here.
“Lille are saying they’ve got significant proposals from La Liga, Serie A – I believe Napoli are one of the clubs with an interest in Pepe – and England.
“Arsenal and Everton have enquired, Manchester United I told you back in May had made their first enquiry direct to the club about Pepe to ask what his salary would be, what kind of money Lille would want for the player.”
On United’s current stance, Castles continued: “They have advanced that interest in the last week and been involved in further discussions with Lille over a transfer.
“It would make a lot of sense for Manchester United. We’ve seen the way Ole Gunnar Solskjaer is setting his team up for next season and it’s very much focused on pace in attack.
“Pepe can play anywhere across the forward line and is left-footed – they don’t have a left-footed forward in their senior squad at present.
“The question will be, can they convince the player to come if Lille get an offer for their asking price from more than one club, and some of the other clubs are able to offer Champions League football?”