The Spain attacker was convinced of a big future under new Real coach Carlo Ancelotti, but is rethinking that stand after a difficult start to the season.
Asensio is now reconsidering his options and could seek a move away in January.
A source said: "Asensio's environment has said that his future depends on Ancelotti. The injury and his tumultuous summer have meant that this summer he was able to leave.
"There is a fundamental fact. It is the first summer in which he has been able to leave. A serious offer has arrived and he has stayed, but Asensio has always (previously) been untouchable."
Asensio was left on the bench for Sunday's victory at Valencia, though is expected in tonight's squad for Real Madrid's LaLiga clash with Real Mallorca. Asensio came through the youth system at Mallorca before being snapped up by Espanyol. He eventually made the move to Real Madrid in 2017.
So far this season, Asensio has made five senior appearances, largely off the bench. The midfielder has a deal with Real Madrid to 2023.
- updated September 22
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