Both clubs have been heavily linked with the winger who is reportedly ready to leave France this summer.
Pepe enjoyed a productive campaign in France last year, scoring 23 times and providing 11 assists.
The Ivory Coast international is currently on holiday after representing his country at the African Cup of Nations.
But he could have a huge decision to make about his future when he returns.
According to L’Equipe, Arsenal, Liverpool and Inter have all made offers for Pepe.
Lille are waiting to hear what the 24-year-old’s preferred destination is before opening discussions.
Napoli and Atletico Madrid are also interested in signing him.
Lille would demand around €80m (£72m) including bonuses.
“Arsenal, Liverpool and Inter have all made offers for Pepe.”
Arsenal have just £45m to spend this summer and additional funds will have to be raised by player sales.
The Gunners have already been priced out of a move for Wilfried Zaha, so it’s unlikely they’d be able to afford Pepe.
But Italian outlet FC Inter News claim United are plotting a £62m bid for Pierre-Emerick Aubameyang.
And should the deal go through, Emery could target Pepe to provide a threat out wide.
Meanwhile, Liverpool are not expected to splash out this summer.
Reports have suggested Jurgen Klopp will only look to replace any fringe players that leave the club.
But if Pepe were to become available, the Reds may be tempted to bring him in to provide competition for Mohamed Salah and Sadio Mane.
And Pepe could be joined out of the Lille exit door by defender Adama Soumaoro, who is interesting Crystal Palace.