Liverpool agreed to a deal with Leipzig in August of last year worth £66.4m (€75m) for Keita to come to Anfield on July 1, 2018.
Sources have told Sky Germany that Leipzig are unwilling to let Keita go in January, even for an upgraded fee.
Liverpool have money to spend after selling Philippe Coutinho to Barcelona, who made his move to the Nou Camp on Monday, for a fee agreed to be £146m.
Keita had a release clause in his deal at Leipzig which meant he would have been open this summer for £48m, but the Reds paid a bonus to secure the player 12 months early, amid interest from Barcelona.
The 22-year-old has made 13 features for Leipzig in this season, scoring twice and adding one assist.
Last season, the Guinea international scored eight goals and added seven assists in 31 Bundesliga appearances.
Klopp is ready to move for long-term targets if there is a realistic chance he can get them now. But with Alex Oxlade-Chamberlain settling in at Anfield and Adam Lallana back from injury, he will not rush, nor overpay just because the club is flush with cash.