The France international's contract is now running down and due to expire in June.
Liverpool and Konate's camp have engaged in talks over a new deal for the past 12 months, but negotiations have stumbled over the player's demands.
Konate is currently earning around £150,000-a-week and he has been demanding those terms be improved to place him among the club's highest earners like Florian Wirtz and Mohamed Salah.
Further complicating things for Liverpool has been interest arriving from across Europe, with Real Madrid, PSG and Inter Milan expressing interest.
However, it's understood a breakthrough has been found, with L'Equipe reporting Konate's representatives have now reached a broad agreement with Liverpool management over a new, long-term contract.
For his part, Konate is hoping to have it all settled before the season concludes.
Reds manager Arne Slot stated in February: "We are in talks with him so that tells you what we want.
"It’s clear we would like him to stay but negotiations are ongoing so let’s see where that ends. We wouldn’t be in negotiations if we didn’t want him to stay."