It is understood Bayern Munich are plotting a January move for the striker.
According to Bild, the German club are now in pole position to land Pepe when the transfer window reopens.
They want to speed up talks in order to fend off potential competition for the Ivorian.
With both Arjen Robben and Frank Ribery, aged 34 and 35 respectively, into double figures for appearances this season, the Bavarian giants need some fresh legs on the wing.
Pepe, 23, has impressed so far this season, notching up six goals and six assists for Ligue 1 side Lille in just nine appearances.
He bagged a hat-trick in the 3-2 away win over Amiens last month and has three goals in his eight international caps for Ivory Coast.
Arsenal were keen to land Pepe next year with a £30million bid for the star, who signed a five-year deal at the club in 2017.
Manchester United joined the race for the highly-sought-after attacker towards the end of September.
Meanwhile Southampton and French side Lyon both had bids rejected by Lille in the summer.
But it appears he could be tempted by a move to the Bundesliga to continue his development.
And Bayern are in need of something different after a poor start to the new campaign.
Niko Kovac is already under pressure with just four wins from seven this season and a heavy 3-0 home defeat to Borussia Monchengladbach did not help.
French trio Zinedine Zidane, Arsene Wenger and Laurent Blanc are on a list of potential candidates to replace the Croat.
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