Spanish outlet Fichajes (via the Telegraph) reported the Spaniard may be looking for a move away from the Santiago Bernabeu as he has lost his first-team spot to Danilo. However, the Blues would have "to shell out for his services" as he only signed a new five-year deal last year.
Carvajal has only made eight Liga starts this season after he made 25 in the 2014-15 campaign—an indication he has fallen down the pecking order at Real, per WhoScored.com.
However, the 24-year-old is still an effective and exciting defender and could be a major asset in the Chelsea squad, as the Blues are currently overreliant on the declining Branislav Ivanovic at right-back.
Though the Serbia international has stepped up his performances after a shocking start to the season, he is still performing far below his best.
At 31 years old, there is little chance of Ivanovic returning to his previous form, and Carvajal would be an excellent long-term addition to the Blues defence.
However, there is little likelihood Los Blancos will be prepared to let him go, as not only does Carvajal provide valuable cover at full-back, but his recent performances have shown he deserves a consistent spot back in the first team under Zinedine Zidane, per Spanish football writer David Cartlidge:
David Cartlidge @davidjaca
Carvajal has matured so well. He's the one that looks like a €30m full-back. Difference in team with him, impetus and security he provides.2016-1-24 21:31:12
Elsewhere, imminent Blues signing Alexandre Pato could be set for a return to former club AC Milan in the summer following his prospective six-month loan spell at Stamford Bridge.
According to La Stampa (via Gazzetta World), the Rossoneri could go back in for the striker after his spell at Chelsea as he proved back in Brazil with Sao Paulo and Corinthians he may have gotten over the fitness issues that plagued him when he was at the San Siro.
Per BBC Sport, Pato's move to Chelsea until the summer is all but done, although the west London club has yet to confirm it.
Gazzetta World's report indicated Chelsea will have the option to sign 26-year-old Brazilian permanently at the end of his spell.
As a result, they should be protected against Milan swooping in and taking him off them in the summer, but if Chelsea decide not to extend his deal, the Italian giants could return for the player who excelled for them in fits and starts between 2007 and 2012.
However, per Bleacher Report's Dean Jones, the Blues could be looking to hold onto as many players as possible as they may soon be under a transfer ban:
According to the Daily Mail's Neil Ashton, the Blues have been asked by FIFA to state the number of games Bertrand Traore played for them as the governing body continues its investigation into a suspected breach of rules over the international transfer of minors.
Traore appeared on the Chelsea teamsheet for an U18 friendly against Arsenal at the Emirates Stadium in 2011, a game he should not have appeared in as he was only 16 at the time—a minor—and shouldn't have been registered.
Per Ashton, the Burkina Faso international is thought to have played in further matches for the Blues behind closed doors, and Chelsea are running the risk of being sanctioned as Atletico Madrid, Barcelona and Real Madrid have all been in the past.