The 21-year-old attacker spent the first half of the season with RB Leipzig in Germany, but saw his temporary deal cut short by Liverpool amid a lack of game time.
Carvalho started just once in the Bundesliga during his six-month stay at Leipzig and found game time hard to come by under manager Marco Rose.
The former Fulham star has since returned to Anfield, with the Reds now weighing up what to do with the young talent in the second half of 2023/24.
A host of clubs have been touted as potentially interested in a short-term move for Carvalho, while there is also the option that he could stay as a squad player at Liverpool for the fixture-heavy run-in as the Reds compete on multiple fronts.
Leicester, Southampton and Fulham have all previously been suggested as suitors. However, more sensationally Everton have been named as one club "asked for info" about the Portugal Under-21 international.
Journalist Rudy Galetti claims that the Liverpool's city rivals have made an enquiry about Carvalho, along with Wolves and Leicester, with the player and his club set to take a decision on the next steps.
The last players to transfer between Liverpool and Everton
A switch (albeit a temporary one) to Everton would be a remarkable turn of events, with transfers between the two rivals a rarity in modern times.
The last player to move from the Reds to the Toffees was defender Gary Ablett way back in January 1992, with Peter Beardsley taking the same career path the summer before.
Two players have switched between the clubs since then, but both moved in the other direction from Goodison Park to Anfield.
Nick Barmby crossed the divide in July 2000 and played a key role in the Reds' treble-winning season in 2000/01. Meanwhile, Abel Xavier was the last player to transfer directly between the clubs when he left Everton for Liverpool in January of 2002.
Defender Conor Cody has also played for both rivals since – but signed for Everton in 2022, almost ten years on from his sole Premier League outing for Liverpool.