And without an adequate back-up he has gone without a striker recently when Morata has been absent. This has left the Chelsea manager with limited options up front.
Antonio Conte has reportedly had a target man in his sights for some time, and, after failing to land Fernando Llorente over the summer, the Italian is now purportedly considering the nine-times-capped England global.
The 28-year-old has had another injury-plagued season, starting just eight games.
Moyes side earned a point at Wembley
Carroll, who scored twice against West Bromwich Albion at the start of the week, featured as a late substitute for the Hammers in last night's 1-1 draw against Tottenham.
In a Chelsea side composed of majestically gifted footballers, the 6 ft 4 in Carroll would offer an injection of height and power that few had really considered to be of great importance to Conte.
"It's been very frustrating". It's hard when you're coming on for the last 20 minutes.
"I give my opinion to the club but the club then decide the players we can buy or sell", Conte added.
"The club signs the players, not me".