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BBC Sport understands that negotiations have stopped and Konate will become the latest player to leave the club on a free this summer after Andy Robertson and Mohamed Salah.
Last year, defender Trent Alexander-Arnold joined Real Madrid a month before his contract expired after the Spanish club paid a fee to release him early to play in the Club World Cup.
Captain Virgil van Dijk's current deal expires next summer, while the club failed to sign Marc Guehi on deadline day last September, with the England player joining Manchester City in January.
Liverpool are confident they have sufficient depth at centre-half after recruiting Giovanni Leoni last summer and the arrival of £60m Jeremy Jacquet, 20, this summer.
But it does leave Van Dijk, 34, as their only experienced central defender.
Frenchman Jacquet, who turns 21 in July, played 21 games for Rennes last season but missed the last four months with a shoulder injury.
Leoni, 19, was ruled out for a year after after tearing his anterior cruciate ligament in September, a month after joining the Reds from Parma for a fee of £26m plus add-ons.
The belief within Liverpool is that other areas - like replacing Mohamed Salah and filling the gap after Hugo Ekitike's injury - are priorities, rather than agreeing to an expensive renewal for Konate.
