Carroll has not featured since April due to a groin problem and it has been suggested that the Hammers are ready to offload their former club-record signing - who has made only 52 starts since 2013 - following the arrivals of Javier Hernandez and Marko Arnautovic.
The Sun reported over the weekend that Newcastle, who have failed to land a number of priority targets amid well-documented friction between manager Rafael Benitez and owner Mike Ashley, were considering a shock move to bring the former England international and Gateshead native back to St James' Park.
The Magpies signed Joselu from Stoke City in a £5m ($6.4m) deal last week to strengthen a meagre strike-force consisting of only Dwight Gayle and Aleksandar Mitrovic but it could be that they seek to make further attacking additions before 31 August.
Quizzed on Newcastle's supposed interest in re-signing Carroll six-and-a-half years after selling him to Liverpool for £35m, a dismissive Bilic, appearing at a press conference held prior to Wednesday night's (23 August) Carabao Cup second-round trip to League Two side Cheltenham Town, told reporters: "He's always a very, very interesting character to be in the papers for whatever reasons.
"I think this is because we play Newcastle, he's from Newcastle, he's on the verge of coming back to the team. You have to fill the papers and I think that's the case, as far as I'm aware."
Bilic revealed that Carroll was now into his second week of full training, but there remains a "big question mark" as to whether or not he will be available to play against Newcastle on Saturday. If the 28-year-old is absent from that trip to Tyneside - West Ham will make a final decision on Friday - then he is definitely expected to feature immediately after the international break.
West Ham's previously cloudy injury picture now looks much clearer, with Bilic, who plans to take a strong squad to Whaddon Road as he looks to build confidence following demoralising back-to-back defeats that have seen his team ship seven goals, confirming that all apart from Winston Reid trained without restrictions on Tuesday. Such a list includes the likes of Manuel Lanzini and Cheikhou Kouyate.
Influential centre-back Reid was worryingly stretchered off during the warm-up before last weekend's eventful 3-2 defeat at Southampton after injuring a calf muscle, although his manager was keen to stress that the problem was not a bad one and that he should be ready to play against high-flying Huddersfield Town on 11 September.
Bilic was notably coy on transfers heading into the final week of the window, admitting that further comings and goings were possible. However, he would not be drawn on specific links to Sporting Lisbon midfielder William Carvalho or a potential exit for £10.2m January arrival Robert Snodgrass.