Speculation was rife last week that Big Sam is lining up a January bid for the homegrown central defender, as reported by the Mirror.
Allardyce was appointed Black Cats boss this week on a two-year deal after a short break following his departure from the Hammers back in May.
According to a report in the Mirror, Big Sam has been guaranteed a transfer war chest, with chairman Ellis Short desperate to avoid relegation.
Bolstering the squad and shoring up the Wearsiders' brittle defence is unsurprisingly Allardyce's key priority, and the Mirror claims homegrown Hammer Tomkins, 26, is his number one target.
'He won't be going anywhere'
But while Allardyce knows Tomkins well from his four-year spell in the Upton Park hotseat, he has been told in no uncertain terms not to bother wasting his time with a bid by Sullivan in his latest column on West Ham's official website.
"There has been speculation in the media surrounding the future of our talented defender James Tomkins," Sullivan told whufc.com.
"James is a West Ham United player and has four-and-a-half years left on the contract.
"He loves the club.
"He won’t be going anywhere."
From the horse's mouth but mega offer could change everything
So there you have it, word from the top that Tomkins won't be going anywhere.
It is a slightly unusual step for Sullivan to take the time out to deny what seemed some pretty routine transfer speculation, particularly given Tomkins faces a return to bench duty when Angelo Ogbonna is fully fit again.
But you get the sense Sullivan has made an exception, given who Sunderland's new boss is.
We all know money talks in football, though, and if Sunderland stump up enough cash, it could change everything.
But whether Tomkins would want to swap playing out the last season at West Ham's famous Boleyn Ground home of 112 years for a relegation battle with the Black Cats is another matter entirely.