Konate will leave Liverpool when his contract expires this summer after negotiations over his wages collapsed with the club earlier this month.
The French defender won the Premier League, the FA Cup, two Carabao Cups and the FA Community Shield during his 183 appearances on Merseyside.
As summer approaches, he will have his pick of the clubs over the coming months, including Spanish giants Real Madrid, with whom Fabrizio Romano reports he has reached a verbal agreement.
"BREAKING: Real Madrid have verbal agreement in place with Ibrahima Konate!
"Four-year deal to be signed, depending on if/when Florentino Perez is re-elected.
"Here we go, expected right after. Konate put Saudi bids on hold for Real Madrid."
The move would see Konate reunite with Trent Alexander-Arnold, a player who has infamously had a tricky time in the Spanish capital since his move last summer, also at the end of his contract.
Konate has previously expressed a desire to represent his boyhood club, Paris Saint-Germain, but no interest from the European champions has been shown as of yet, as he creeps closer to LaLiga.
As he leaves the Reds, Konate expressed his gratitude to the fans but revealed it is time for him to depart.
"From the bottom of my heart, thank you for everything. I love you all and will carry Liverpool with me wherever I go. This isn't an easy goodbye, but it's time for a new challenge and a new chapter."