Pepe has attracted interest from across Europe this year after spearheading Lille's impressive season of finishing second behind PSG.
As a result, his tally of 23 goals and 12 assists has prompted the Ligue 1 cliub to slap an €80 million price tag on his head.
L'Equipe claim that Inter Milan made the first move for Pepe by bidding €70m.
But they also say that Liverpool have 'moved up a gear' and aren't far off Lille's valuation in their latest proposal.
Pepe is believed to favour a move to La Liga and wants guaranteed playing time.
Both factors might play against Liverpool, as the report points out, but that hasn't deterred Jurgen Klopp from making a move.
Liverpool have confirmed that Daniel Sturridge will be leaving the club as his contract expires at the end of the month, and are now looking to replace the Englishman with more quality in the final third.
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