After his contract at Manchester United was cancelled last month, Ronaldo is free to join another club on 1 January.
Saudi Arabian club Al-Nassr have made a big-money offer for the 37-year-old forward, although it is understood he has other options.
Ronaldo trained alone at Real's Valdebebas base and there is no option of a return to the Spanish club.
Ronaldo's second spell at Manchester United was brought to a premature end after a controversial interview in which the five-time Ballon d'Or winner criticised the club and said he had “no respect” for manager Erik ten Hag.
The Portuguese Football Federation denied that Ronaldo threatened to leave the squad after being told he would not start the last-16 game against Switzerland, and his replacement Goncalo Ramos went on to score a hat-trick in a 6-1 win.
The tournament in Qatar was expected to be Ronaldo's last chance to win a World Cup, and he came off the bench against Switzerland and Morocco but could not prevent the African side knocking out Portugal.
That 1-0 upset in the quarter-finals could be Ronaldo's last World Cup game and he said on social media: “The dream was nice while it lasted”.
In asking Real to use their training facility rather than take a break following Portugal's early exit, Ronaldo has given a clear indication he is ready to join a new club at the earliest opportunity.
However, that will not be Real, whose all-time leading scorer is Ronaldo with 450 goals in 438 games, during time which he won the Champions League four times and La Liga twice.
Ronaldo made a then-world-record move to Real from Manchester United in 2009 and left for Juventus in 2018.