"Talks between the clubs could conclude on Friday about a deal that may amount to 60m euros", reported the BBC.
Fekir has also drawn interest from Chelsea and Bayern Munich, but Liverpool have been aggressive in chasing his signature.
The Lyon captain is the Reds' primary target as they seek to add creativity to their attacking midfield options.
Liverpool's pursuit of the France global involves dealing with Aulas, who is known to be an extremely tough negotiator, and while a source said on Thursday that a fee has yet to be agreed, discussions have taken a positive turn and there is optimism growing on Merseyside that Liverpool could secure a deal prior to France flying out to Russian Federation.
Weary Alexander Zverev out as Dominic Theim reaches last four
Wozniacki did however make the quarterfinals a year ago in Paris prior to losing to the eventual champion Jelena Ostapenko. Putintseva matched her best Roland Garros performance by reaching the quarter-finals for the first time since 2016.
Les Bleus will take on the United States this weekend, in their final World Cup tune-up.
Red Bull Leipzig midfielder Naby Keita is already en route to Anfield having agreed a deal to leave the Bundesliga side last summer.
Liverpool are also now being linked with Stoke City's Xherdan Shaqiri and The Independent state that the Reds "are primed to seal at least one new signing in the next 24 hours".
France's squad fly out to Russian Federation after their final friendly against US in Lyon on Saturday.
Reports on Wednesday also suggested that Liverpool were giving serious thought to activating the relegation release clause in Xherdan Shaqiri's contract at Stoke City.