They'll loan Ings for a year before automatically buying him for £18 million next July, with two million pounds of add-on's and a 20% sell-on fee set to benefit Liverpool.
Klopp leads tributes to departing Ings
Despite his bubbly personality seeing him become a hit around Melwood, Ings' on-pitch fortunes at Anfield often left him frustrated, with two serious knee injuries affording him just 25 appearances and four goals since joining from Burnley in 2015.
Falling behind Roberto Firmino, Daniel Sturridge and Dominic Solanke in the pecking order left him with little choice but to angle for a move this summer, Klopp full of praise for the 26-year-old after granting his wish.
"It is a very strange feeling saying goodbye to Danny," said Klopp.
"The biggest emotion is how much we will miss him. He is such an outstanding boy. Honestly if you could bottle character & spirit, his would sell worldwide."
Ings will offer an alternative prong to Saints' frontline, providing competition for Manolo Gabbiadini and Charlie Austin.
A South-Coast boy himself, Ings' return 'home' will give Saints a good opportunity to improve in the goalscoring department, as they struck just 37 times in the league last season.