The Daily Telegraph says Conte is also keen on signing Red Star Belgrade's Richmond Boakye and Crystal Palace's Christian Benteke, although Carroll, 29, is seen as the most attainable before the January transfer window closes.
Chelsea have played out three successive goalless draws – against Norwich, Arsenal and Leicester, respectively – and are on a four-game winless run in all competitions.
The Telegraph says Conte is concerned over the form of Alvaro Morata, who has not scored in his last five games for the club, and Eden Hazard.
The Blues boss also has Michy Batshuayi at his disposal, but the Belgian striker has been told he can leave the club if a replacement is signed.
Chelsea were also linked with a move for Moussa Dembele earlier this month, but their interest has since cooled over Celtic's demands for a £40m ($55m) transfer fee.
Conte expressed admiration for Alexis Sanchez last week, but he is resigned to the fact that the Arsenal forward is likely to join one of Manchester City or Manchester United.
"I have great admiration for great players, I think Alexis Sanchez is one of these players," Conte was quoted as saying by Goal.
"I don't know [where Sanchez will go]. You have to ask him which is his choice, or maybe ask Arsenal because Arsenal has to decide his future before June. For sure, his future will be in a great club."
Chelsea signed Ross Barkley from Everton in a £15m deal earlier this month, with the midfielder expected to make his Blues debut against Norwich in their FA Cup third-round replay at Stamford Bridge on 17 January.
The Blues are fourth on the Premier League table, level on points with second-placed Manchester United.