The Dutch midfielder left Anfield after five years with the club when his contract expired in June, and he has since joined Paris Saint-Germain on a free transfer.
Klopp was keen to keep hold of Wijnaldum as he was still part of his plans going forward, but according to The Athletic, FSG were unwilling to hand the 30-year-old a new deal.
The German boss is believed to have an input on who stays and who is recruited at Liverpool, though the Fenway Sport Group – who bought the club in October 2010 – have the final say regarding financial matters.
As well as Wijnaldum, captain Jordan Henderson could also find himself in a similar position, as talks to extend the 31-year-old's contract have stalled this summer.
Liverpool's owners are concerned by the age of several first-team members and they are planning to freshen up the squad in 2022, with the primary focus on bringing in younger players.
The Reds have already made one signing this summer, recruiting 22-year-old defender Ibrahima Konate from RB Leipzig for around £36m. body check tags ::