There is competition from a number of clubs, including Chelsea, for Fekir's signature, although the Reds are confident that they can see off other potential suitors and get the deal over the line.
Indeed, Jurgen Klopp's side are keen to make a number of key signings this summer, and have identified the France international as one of those.
Fekir, 24, has been heavily linked with a move to Anfield in the French media, with reports earlier this month claiming a deal has already been struck.
That is not the case at this stage, but Liverpool are now ready to open negotiations with Lyon following their Champions League defeat to Real Madrid on Saturday. Goal understands Fekir would welcome a move to Merseyside, though he told TV1 on Sunday that the deal was "not close".
Liverpool are prepared to spend big this summer as they seek to strengthen a squad which came close to glory but is lacking in several areas. Fekir, a creative, goalscoring attacking midfielder, is seen as the man to replace Philippe Coutinho, who left for Barcelona in January.
"Will I sign anywhere before the World Cup? Honestly, I don't know," he said after netting in France's 2-0 friendly win over Ireland.
Naby Keita, the Guinea midfielder, will join from RB Leipzig on July 1, while Liverpool moved unexpectedly on Monday to sign Fabinho from Monaco for €45m. He is the defensive midfielder they craved.
A multi-functional forward is also wanted, with Klopp aware that the team has been heavily reliant on Mohamed Salah, Roberto Firmino and Sadio Mane this season – a point illustrated during the Champions League final, where the Reds struggled badly after Salah was forced off through injury.
Christian Pulisic, the Borussia Dortmund wide player, is a favourite of Klopp’s and Liverpool will explore the possibility of signing the American this summer.
Elsewhere, there will be renewed calls for a new goalkeeper following Loris Karius’ mistakes against Real Madrid.
Liverpool’s feeling is that they will only spend big on a ‘keeper if they are sure they are a clear upgrade on Karius, who still retains the support of the manager.
Alisson Becker, the Roma No.1, would be seen as one such upgrade, but Liverpool have been put off by the Italian club’s asking price, which is only likely to rise given Karius’ misfortune in Kiev. Goal understands Jack Butland, meanwhile, is nowhere near as highly-rated as Alisson.
The Reds must also address the future of Simon Mignolet, their own second-choice shot-stopper. The Belgian is reluctant to spend another season on the bench, and will ask the club for clarification over his role, with a number of suitors interested. Danny Ward, the No.3 goalkeeper, may also wish to leave in search of first-team football.
Decisions must also be made on a raft of other players, most of whom spent last season out on loan. Daniel Sturridge will be sold, while Divock Origi may get the chance to prove himself on the pre-season tour of the USA.
Harry Wilson will have no shortage of offers after a good five months playing at Hull City, and Premier League newboys Fulham are among those interested in midfielder Marko Grujic. Lazar Markovic, Ovie Ejaria, Sheyi Ojo, Ryan Kent, Pedro Chirivella, Jon Flanagan and Connor Randall could all depart too. Allan Rodrigues de Souza and Taiwo Awoniyi will be loaned out.
Finally, doubts remain over teenage star Rhian Brewster, with Borussia Monchengladbach hoping to sign the England Under-17 World Cup winner. Brewster is yet to pen a professional contract with Liverpool, who are desperate to keep the striker.
The club remain in discussions with the 18-year-old and his representatives, and are willing to report both Gladbach and RB Leipzig for what they believe to be illegal approaches to the player. A friendly with Gladbach, scheduled for early August at Anfield, has been cancelled as a result of the situation.