In January, Liverpool allowed Philippe Coutinho to leave in a big-money switch to Barcelona and it has left the Merseyside giants with money to spend in the summer transfer window.
However, according to AS, Jurgen Klopp has missed out on trying to tempt Real into the sale of one of their biggest prospects ahead of next season.
Earlier this week, it was claimed that Real had rebuffed two bids of £132m, and the most recent report is suggesting that Liverpool were one of the clubs to make an approach.
During the current campaign, Asensio has contributed seven goals and seven assists in 43 appearances in La Liga and the Champions League.
The Spain international still has five years remaining on his contract at the Bernabeu.