The Real Madrid winger is available in the transfer window, as he is surplus to requirements at the Santiago Bernabéu stadium club.
According to The Mirror, Asensio has no way back at Real, and is hoping to earn a move to a top club this summer.
The source states that Klopp could be ready to offer the 26-year-old a new challenge in the Premier League.
Real would want around £35 million for Asensio, but the stumbling block may be convincing the player.
Asensio may not feel he would get much more game time at Liverpool than he does at Real.