The 26-year-old France international, who joined from RB Leipzig in 2021 for a fee of 36 million pounds ($49 million), had been linked in the media with a move to Spanish giants Real Madrid.
Konate has been an integral part of Arne Slot's side, forming a formidable defensive partnership with captain Virgil van Dijk. During his time at Anfield, he has won the Premier League, two League Cups and the FA Cup, and finished runner-up in the Champions League.
"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," Konate told reporters after Liverpool's last-gasp 2-1 derby win over Everton on Sunday.
"I think everyone wished for that as soon as possible, but we are in a good way.
"For sure, there is a big chance that I'm here next season. This is what I've always wanted.
"I'm waiting to sort the contract, but when everything is sorted you will have to ask Richard Hughes (Liverpool' sporting director) what I said to him in September and November, before everyone talked about everything. He's going to say something to make everyone quiet."
The current campaign is set to end without silverware for the defender, with Liverpool eliminated from the Champions League, FA Cup and League Cup, and sitting fifth in the Premier League with five matches remaining.
"It's been a bad season for the standards of Liverpool. If we finish in the top four, we won't be happy, for sure," Konate added. "It's such an amazing club and an amazing family. This club means so much to me."