The Lyon ace has been heavily linked with a move to Anfield in recent weeks.
Fekir, 24, could become Liverpool’s third summer signing, with the midfielder following Fabinho and Naby Keita to the Reds.
And according to The Gambling Times, the France international has all but agreed terms with Liverpool.
They reckon the Reds are hopeful of completing the deal before the World Cup starts next week.
The Gambling Times add that Liverpool are currently refusing to pay over £50m for Fekir, while Lyon want £60m.
The clubs could meet in the middle, with the Reds forking out around £55m for the France star this summer.
Lyon president Jean-Michel Aulas has accepted that Fekir sees his future at Liverpool under Klopp.
Fekir has been vocal about his future in recent weeks, with the midfielder revealing he may make a decision after the World Cup.
“If Fabinho will be my future team-mate? I will discuss about my future after the World Cup, not before,” he said.
“If I will sign somewhere before the World Cup? Honestly, I don’t know.”
Liverpool are looking to spend big this summer to mount a Premier League title charge next season.