The 26-year-old France international, who has been a key figure on Merseyside, is in the final months of his current deal, which is set to expire at the end of the season.
That situation had raised the prospect of the defender leaving on a free transfer this summer, particularly amid reported interest from Real Madrid.
However, negotiations between the club and the player are now understood to be nearing completion, with sporting director Richard Hughes playing a central role in advancing talks.
Konate himself has indicated confidence that an agreement will soon be finalised, stating that he expects to sign a new contract in the near future.
"My future? To be fair, there are many things people have said but for a long time we have spoken with the club and we are close to an agreement," said Konate.
"I think everyone wished for that for as long as possible but we are in a good way. For sure there is a big chance I'm here next season."
The development is viewed internally as a significant positive for Liverpool, effectively bringing an end to months of uncertainty surrounding one of their most important defensive assets.
Retaining Konate would provide continuity and stability at the heart of the defence, reinforcing the club's long-term planning.
Liverpool midfielder Dominik Szoboszlai has played down concerns over his contract situation, insisting he remains calm despite a lack of progress in ongoing talks with the club.
Arne Slot has praised his Liverpool side's mentality and character after a dramatic 2-1 victory over Everton in a tense Merseyside derby decided deep into stoppage time on Sunday.
Liverpool need a very late Virgil van Dijk goal to claim a 2-1 victory over Everton in their Premier League clash at the Hill Dickinson Stadium on Sunday.
Jarell Quansah has admitted he had "started to love football again" after making a fresh start with Bayer Leverkusen in the Bundesliga.
