The 22-year-old joined the Emirates club in January on a deal until the end of the season, with no option to make the move permanent.
Odegaard has made a good impression in his nine Premier League appearances so far, the last of which came in a 3-0 defeat to Liverpool on Saturday.
Reds boss Jurgen Klopp spoke to and embraced the Norwegian after the game, and AS believes it could have been more than an innocent exchange of words.
The German manager was interested in Odegaard when he first took the Anfield job in 2015, and the player is open to a move to Merseyside.
Arsenal want to make the loan permanent this summer but could struggle to afford a deal, leaving Liverpool with the opportunity to offer him a way of staying in England.
Odegaard’s contract in Madrid has another two years left to run, but if he isn’t offered more regular game time on his return to the Spanish capital, he will ask to leave.
The playmaker has scored two goals in 13 appearances for Arsenal in all competitions, helping them reach the Europa League quarter-finals, where the Gunners face Slavia Prague on Thursday.
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