"Talks between the clubs could conclude on Friday about a deal that may amount to 60m euros", reported the BBC.
Fekir is due to undergo a Liverpool medical at Clairefontaine, the France national team's training base, before signing terms on Friday.
But L'Equipe claim that an agreement has been reached and Fekir's transfer is likely to be announced before the weekend.
"The whole thing will certainly not be completed during the World Cup, and I don't think anything will change before it".
The 28-year-old has only featured in 18 matches for the Saints in the last two seasons and could join a long-list of former Southampton players who have joined the Merseyside club.
Fekir has also drawn interest from Chelsea and Bayern Munich, but Liverpool have been aggressive in chasing his signature. Didier Deschamps' side play their final World Cup warm-up game against the U.S. in Lyon on Saturday and the manager had instructed his squad to settle any outstanding transfer business before flying to Russian Federation.
The Independent report that Aulas believes that Liverpool will have to calculate the potential of summer performance into their offer.
Jurgen Klopp is also weighing up whether to trigger Xherdan Shaqiri's £12m release clause following Stoke City's relegation from the Premier League.