"Talks between the clubs could conclude on Friday about a deal that may amount to 60m euros", reported the BBC.
But Lyon President Jean-Michel Aulas earlier said the deal would NOT go through until after the World Cup.
Fekir will be given permission to leave France's World Cup training base and could undergo his medical on Merseyside on Friday.
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And the Mirror now say the Reds are confident the 24-year-old will be confirmed as a Liverpool player before the weekend. The case won't certainly be resolved during the World Cup and I do not think that there will be changes before (the WC). "We haven't set a price tag because we haven't discussed this [Fekir's transfer] yet".
The Independent report that Aulas believes that Liverpool will have to calculate the potential of summer performance into their offer.
Bayern Munich and Atletico Madrid are also reportedly interested in the French worldwide - but Liverpool are now in the driving seat.
A fourth transfer could soon be on the cards with reports claiming a £12m deal for Stoke's Xherdan Shaqiri could be in the pipeline.