With Davies out of contract in 2025 and supposedly at odds with Bayern over an extension, speculation over a summer move to the Bernabeu for the Canadian has been hotting up in recent months.
There is yet to be a resolution to Davies's contract impasse, meaning that Bayern could be pressured to sell the left-back in January if they fail to stump up an offer that is deemed satisfactory by his entourage.
Bayern boss Thomas Tuchel is keen to retain Davies's services, though, and even if the left-back decides that his future lies elsewhere, Chelsea could also try to prise him away from Bavaria.
Should Real be forced to concede defeat in their efforts to sign Davies, Fichajes claims that Leverkusen hotshot Grimaldo has emerged as a backup option amid his explosive first campaign in the Bundesliga.
The 28-year-old had already made a name for himself at Benfica in a seven-year spell between 2016 and 2023, having been allowed to depart the Barcelona academy for a mere €2.1m (£1.8m) after eight years in their La Masia academy.
However, Grimaldo has exceeded expectations in his left wing-back role under Xabi Alonso, already registering an eye-catching nine goals and 10 assists in 26 appearances in the 2023-24 campaign.
The Spain international has seven goals and nine assists under his belt from 18 Bundesliga contests, with only Bayern Munich's Leroy Sane (10) setting up more efforts in the current German top-flight season.
While he is without a Bundesliga goal since the end of November, Grimaldo has provided assists in each of Leverkusen's last three top-flight matches, including two for Nathan Tella and Piero Hincapie in Saturday's absorbing 3-2 win over RB Leipzig.
With the Spaniard bombing down the left flank, Leverkusen have made a sensational start to the top-flight campaign with 48 points from their first 18 matches, allowing them to open up a seven-point lead over perennial champions Bayern Munich at the top of the table.
Leverkusen have Grimaldo under contract until 2027 and are therefore under little pressure to sell the left-back, but the German club will be braced for summer bids if he continues performing in the same vein.
The report adds that Barcelona would be interested in bringing Grimaldo back to Camp Nou this summer, but Real remain confident of fending off Blaugrana interest should an intense head-to-head tussle ensue this summer.
Since the departure of the legendary Marcelo, Ferland Mendy has attempted to fill the Brazilian's boots, but the former Lyon man has been plagued by injuries since joining Los Blancos in 2019 and is out of contract in 18 months.
Carlo Ancelotti's side recruited ex-youth product Fran Garcia from Rayo Vallecano over the summer, but the 24-year-old has only made the first XI on nine occasions in La Liga this season.