The midfielder only moved to Newcastle from PSV Eindhoven last summer.
Steve McClaren snapped up the 25-year-old for £14.5m – but he lasted just a single season on Tyneside.
The midfielder was unable to keep the Toon in the top flight and has now left following their relegation to the Championship.
Wijnaldum is Klopp’s fifth signing this summer, joining Loris Karius, Sadio Mane, Ragnar Klavan and Alex Manninger.
Joel Matip and Marko Grujic are also new arrivals having signed earlier this year.