Liverpool might have been one of the most prolific attacking outfits in Europe this season, but there is little to no quality beyond the fearsome front trio of Mohamed Salah, Sadio Mane and Roberto Firmino, especially since the departure of Philippe Coutinho to Barcelona in January.
Reports earlier this week indicated that talks have been planned for Fekir.
Negotiations are to speed up from next week on an expected £60 million deal.
The Telegraph claim a breakthrough in the last 24 hours has encouraged Liverpool to such a degree the Reds believe they are the stand-out frontrunners.
Lyon had previously said only a "very, very rich" club could hope to secure Fekir's signature this summer, but their stance has now softened.
The 24-year-old's current deal expires in 2020 and Lyon have told Liverpool a bid of £60million would be enough to complete the transfer.
Lyon president Jean-Michel Aulas recently told SFR Sport that Liverpool were yet to contact the French side about a possible move for Fekir, but ESPN FC has been told the Champions League finalists want to acquire the France worldwide in the summer transfer window.
"I know that Nabil, now that we have qualified for the Champions' League, would really like to stay, so we will discuss things calmly".
"He has magnificently earned the right to go to Russian Federation with the French national team", Aulas said.
This came after reports, originating in France, of Liverpool's interest in the versatile midfielder, with the initial suggestion being the two clubs had already agreed a £62 million fee.
"For the moment, nothing is done".