But the German coach is eager to make further additions with RB Leipzig midfielder Keita reportedly on the manager's radar after a magnificent first season in the Bundesliga.
Leipzig's rise has inevitably resulted in the likes of Keita and Swedish winger Emil Forsberg being linked elsewhere, despite chief executive Oliver Mintzlaff recently declaring that it would be "irresponsible" to sanction the sales of any top performers before a maiden foray into the Champions League next season.
Those comments haven't appeared to deter Liverpool however with The Times now reporting they are ready to test Leipzig's resolve with a club record £70m bid for the Guinea international.
The Reds also remain linked with a move for Gunners midfielder Oxlade-Chamberlain, whose contract at the Emirates is set to expire next summer.
According to the Liverpool Echo, Klopp is keen to bring both Keita and Oxlade-Chamberlain to Anfield in order to bolster his midfield options, with the Arsenal star not viewed as a "back-up" up for the former.
Meanwhile, according to the Independent, Arsenal are "open" to the idea of sanctioning Oxlade-Chamberlain's sale. Chelsea however have also been credited with interest in the 24-year-old, with the Mirror reporting the Premier League champions are keen to take advantage of his contract situation.
Liverpool, who reportedly have a £200m budget this summer, could also be in the market for a left-back. While James Milner provided steady cover in the role last season, the club are still without a natural option in that position with Alberto Moreno out of favour and linked with a move away from Merseyside.