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.
Bayern Munich and Atletico Madrid are also reportedly interested in the French worldwide - but Liverpool are now in the driving seat.
Previous reports suggest that the Reds will pay around £60m for the midfielder.
Earlier in the day, Aulas had said in quotes published on Lyon's website that no deal for Fekir will be completed before or during the World Cup, in which the versatile attacking midfielder will be appearing.
What's more, it looks like the deal could be wrapped up before the World Cup, ensuring Klopp beats any suitors who could potentially take a shine to Fékir should he have a stellar tournament in Russian Federation.
According to the Independent, Lyon have changed their stance and are now prepared to sell Fekir to Liverpool before the World Cup kicks off.
The Independent report that Aulas believes that Liverpool will have to calculate the potential of summer performance into their offer.
Fekir is now gearing up for the start of the World Cup in Russian Federation next week.
A very exciting signing for the Reds.
Could he do the same with Fekir and Shaqiri?