The resurgent Hammers are currently dealing with something of a striker surplus and the quartet of Carroll, Javier Hernandez, Diafra Sakho and Andre Ayew have all been linked with moves away from east London with manager David Moyes tending to favour the likes of record signing Marko Arnautovic, Manuel Lanzini and Michail Antonio in the final third.
Chelsea are believed to be eager to boost their attacking options after a run of three successive goalless draws across all competitions and provide a new alternative to first-choice frontman Alvaro Morata. Michy Batshuayi is close to joining La Liga outfit Sevilla having started just three Premier League fixtures since his £33m ($45.4m) move from Marseille in July 2016.
News of their surprise interest in Carroll first emerged earlier this month and reports suggested this week that they remain keen on a short-term loan deal. Indeed, The Mirror claimed last night that West Ham, who may have been offered a swap deal involving long-term target Batshuayi, fear that the former Newcastle United and Liverpool target man has had his head turned by the prospect of such a switch.
Carroll has made only three starts since Moyes succeeded Slaven Bilic in November and, having opened his account for the season with both goals in the 2-1 defeat of West Bromwich Albion on 2 January, has missed the last two matches with another injury setback.
Quizzed about the 29-year-old's future while standing in for Moyes at a press conference held before Tuesday's (16 January) FA Cup third-round replay against League One promotion contenders Shrewsbury Town, Irvine said: "I certainly hope Andy will be a player here at the end of the month.
"I don't think we have had any offers, in terms of taking him on loan, and I don't think the club would entertain him going out on loan."
Per Sky Sports, he added: "That does not help us and we need as much help as we can get between now and the end of the season, so I don't know why that would be attractive to the club if it was to come up. There has been a lot of talk about it, but not necessarily anybody doing anything about it."
Chelsea are also said to be monitoring Crystal Palace's Christian Benteke, while it is also believed that they have now entered the race for Arsenal contract rebel Alexis Sanchez - a player Antonio Conte previously tried to acquire from Udinese during his successful stint in charge at Juventus.
However, the Mail - who suggest that West Ham value Carroll at the £30m mark, up to £15m more than Chelsea - report that the defending top-flight champions have decided not to follow up their interest in Richmond Boakye after scouting the Red Star Belgrade striker earlier in the season.