The Reds thought they had completed the signing of the France international for a fee in the region of £53 million earlier this month but the deal collapsed over reports there was an issue with the medical.
It is claimed that Liverpool had some concerns with the medical report, and, although they were still happy to push on with the transfer, they were unwilling to pay the transfer fee that had been agreed between the two clubs.
That angered Lyon, who pulled the plug on the deal, and reports claim they are now trying to sell him to rival Premier League clubs for a fee closer to £66 million.
Tottenham and Chelsea are reported to be the two clubs that Lyon have been in touch with, and it remains to be seen whether Mauricio Pochettino would be interested in such a deal.
Fekir is currently at the World Cup with France and came on for the last 20 minutes as they beat Australia 2-1 in their opening group match on Saturday.