Since the turn of the year, the playmaker has been heavily linked with an exit from the Bernabeu due to a number of various factors.
As well has his contract being due to expire in 2023, the 26-year-old has fallen down the pecking order under Carlo Ancelotti.
Although the Spain international still made 31 appearances in La Liga during the most recent campaign, 12 of those outings came from the substitutes' bench.
Arsenal and AC Milan have been among the strongest suitors of Asensio, aware that the player could be available for a cut-price fee.
However, Sport suggests that Liverpool and Newcastle are the teams who are most likely to press forward with a proposal in the coming weeks.
Previous reports have claimed that Asensio wishes to be a regular starter at his next club, something which may eventually rule Liverpool out of contention.
The Merseyside giants have already added Fabio Carvalho and Darwin Nunez to their creative options ahead of 2022-23, with Jurgen Klopp now well stocked for choices in the final third.
While Newcastle are not participating in European football next season, Eddie Howe would regard Asensio as a key man if he were to make the switch to St James' Park.
The report hints that Los Blancos officials have little intention of offering fresh terms to Asensio, who has been at Real since 2014.
That has allegedly resulted in his valuation being reduced to in the region of €35m (£29.6m), an attainable figure for the clubs which have been mentioned.
However, it is claimed that Real may further reduce that valuation if a resolution is not found in the coming weeks, a scenario which may result in any interested teams remaining patient in the race for Asensio's signature.
Real recently took a similar approach to part ways with out-of-favour forward Luka Jovic, who is on the brink of linking up with Fiorentina.