The Evening Standard understands that even though the Blues have yet to make an official approach to sign the England international, Carroll would be keen on the "dream" move to Stamford Bridge.
The Hammers reportedly want £20m (€22.6m) from Chelsea for Carroll to leave on a permanent basis and insist they have not received an approach from Antonio Conte's side.
Carroll has scored just two goals this season, both of which came in the vital 2-1 win over fellow relegation candidates West Bromwich Albion, but manager David Moyes is seemingly unwilling to part with him.
Regardless of his measly goal tally this season, ex-West Ham captain Nigel Reo-Coker hopes Chelsea are thwarted to their bid to sign Carroll this month.
"Andy Carroll is a vital player for West Ham, even though they have picked up a few results lately," he told Sky Sports.
"There's still a long way to go this season and he's a player they need to keep right now ≥ for the team, the system and the way they play.
"On his day he can be a handful for even the best defenders in the Premier League and selling him in the January window is not the right thing to do.
"And, despite what you might get for him, you have to remember what position it could leave your team in. Or have they someone in mind who can fit in right now?
"For West Ham, and where they are right now, Carroll is a player they need to keep in this window."
Should Carroll indeed join Chelsea it is expected to trigger the departure of Michy Batshuayi on loan, though West Ham face competition there from La Liga side Sevilla.
The Spanish club are reportedly willing to pay a £1.3m loan fee and a large portion of his £70,000-a-week wages until the end of the season, but the move hinges on Chelsea finding a replacement.
Belgium international Batshuayi is expected to start the FA Cup third round replay against Norwich City on Wednesday (17 January) but his future in West London is uncertain having made just eight starts this season. Ike Ugbo and Jake Clarke-Salter have also left the club this month.
With Diafra Sakho and Javier Hernandez also among the possible departures from the London Stadium in January, West Ham have laid down provisions in the event they do lose a large chunk of their forward line.
Red Star Belgrade's Richmond Boakye is a target, as is Crystal Palace's Christian Benteke.