With Edouard Mendy having moved to Al-Ahli, Arrizabalaga is currently in line to remain as the number one at Stamford Bridge.
The Spain international featured in the majority of the Blues' friendlies in the United States during the summer and should start against Liverpool in the Premier League on Sunday.
However, at a time when it appears that Arrizabalaga has been content in West London, it was alleged earlier this week that Chelsea were open to offers for the 28-year-old.
Arrizabalaga has just under two years remaining on his contract, while Bayern Munich have been credited as admirers with Thomas Tuchel having previously worked with him during his stint with Chelsea.
Nevertheless, the situation has now changed with Real Madrid in need of a new goalkeeper courtesy of Thibaut Courtois potentially missing the entire season with an ACL injury.
Los Blancos are currently being linked with a wide array of stoppers, including Leeds United's Illan Meslier and Sevilla's Yassine Bounou.
That said, club officials are unlikely to want to shell out a big fee given the assumption that Courtois will eventually return to his number-one role at the back end of 2023-24 or next season.
According to transfer expert Fabrizio Romano, the La Liga giants have made contact with Arrizabalaga's representatives to gauge his level of interest in a move to the Bernabeu.
Arrizabalaga, formerly of Athletic Bilbao, is said to be keen to take the opportunity to return to his homeland with a club the size of Real, whether that be on a permanent or temporary basis.
Sky Sport Deutschland reporter Florian Plettenburg adds that Bayern have been informed that they are not regarded as first choice in the race for Arrizabalaga's signature.
Real are now allegedly keen to find an agreement with Chelsea, with Todd Boehly and Clearlake Capital likely to be open to a deal if it proves beneficial in the long term.
Robert Sanchez has recently been acquired from Brighton & Hove Albion and could theoretically deputise between the sticks against Liverpool this weekend if any progress is made in discussions over Arrizabalaga.