The Reds have been chasing the Germany Under-21 international for some time but the process was held up by complications associated with a buy-back clause Bayern Munich inserted into the deal which took the 20-year-old to Leverkusen for £4.4million 12 months ago.
Can will officially join Liverpool on July 1 when all the relevant paperwork can be completed but the player has already signed a contract with the club.
Can had a 12 million euro (£9.75million) release clause effective for foreign clubs and it is reported Bayern will receive £1.2million as compensation for waiving their buy-back option.
The 20-year-old midfielder follows Southampton striker Rickie Lambert, who completed his £4million move on Monday, to Anfield as manager Brendan Rodgers' summer strengthening continues ahead of a Champions League return next season.
A deal for the German progressed significantly on Wednesday when Leverkusen confirmed Liverpool had made an official bid and it was concluded late on Wednesday night.
"Until Tuesday, it was a rumour," Leverkusen chief executive Michael Schade told German newspaper Bild on Wednesday.
"Now I can confirm that an official offer from Liverpool is received by mail."
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- Bayern Munich