Earlier this week, the German club said they had rejected a bid of over £67m from an unnamed club.
Sky sources understand Liverpool have had three bids rejected for the midfielder.
The Guinea international has a £48m release clause in his contract, although that does not become active until next summer.
Leipzig sporting director Ralph Rangnick is trying to convince Keita to sign a new contract and recently said the club, which has qualified for the Champions League for the first time, would not be selling any of its key players.
Left back Andy Robertson became Liverpool's third signing of the summer when he completed his £10m move from Hull City on Friday, joining Mohamed Salah and Dominic Solanke.
Source: Sky Sports
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