Brendan Rodgers side are said to be close to reaching a deal worth up to €12m for the Germany U21 international who has made a positive impression in his debut season at Leverkusen.
The 20-year-old Can has scored four goals and provided as many assists this season after joining from Bayern Munich in the summer.
It is thought that he is one of a number of players Liverpool are keen to sign as they look to bolster their attacking options ahead of next year's participation in the Champions League.
Can's arrival at Anfield would seem to signal the end of Lucas Leiva's Liverpool career, the Brazilian playing in a similar role to the youngster but also plagued by injuries in recent years.
Leiva has also stoked rumour of his departure with the following tweet after Liverpool's final game of the season: "Yesterday I finished my 7th season with Liverpool.
