Sources have told ESPN that Liverpool's sporting director Michael Edwards has been monitoring Konate for years but that communication between the two clubs has increased since they met in the last round of the Champions League.
A Liverpool scout was watching Konate as early as January 2017 when made his senior debut for Sochaux in the French League Cup against AS Monaco.
Six months after his debut, Konate joined Leipzig on a free transfer after refusing a contract at Sochaux.
Liverpool aren't the only club interested in the France Under-21 international with Chelsea and Manchester United also in contact about the player. Sources have told ESPN, however, that Liverpool are in the lead for the promising centre-back.
However, Leipzig Markus Krosche said Konate, 21, has a big future ahead of him in Germany.
"I think that this is not an option for Ibu," Krosche said of Konate. "He has a long contract with us and he knows what we mean to him and what he means to us.
"He was unlucky last year as he had a long-term injury but this year he is developing well and he is an extremely important player that has a lot of potential.
"We want to help him to develop further, and we play in Europe and have a very good team, so I don't see a reason for him to leave us."
Information from Reuters was used in this report