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.
Barking up the wrong tree?
But while Allardyce knows Tomkins well from his four-year spell in the Upton Park hotseat, is he barking up the wrong tree with a move for Tomkins?
The Academy graduate is one of West Ham's longest serving players, second to only Mark Noble in the current squad.
A boyhood Hammers fan, West Ham showed just how much they value the centre back when they handed him a new five-year contract as recently as February.
And while co-owner David Sullivan recently warned any new signings in January would only be possible if players are sold, as reported by the Evening Standard, it is highly unlikely they would be tempted to let Tomkins go for anything short of an astronomical offer given his importance to Slaven Bilic's side so far this season.
Tomkins the common theme
Although new signing Angelo Ogbonna was preferred to the Basildon-born star in central defence at the start of the season at Arsenal, Tomkins was deployed at right back given that loanee Carl Jenkinson was unable to play against his parent club due to Premier League rules.
And he put in a top performance to help the Hammers to a famous 2-0 win at the Emirates.
Tomkins lost his place in the team to the returning Jenkinson for the next two games, though, and the Hammers were beaten by Leicester City and Bournemouth at home.
A sending off for Jenkinson saw the man with 190 appearances in the claret and blue return to the side at right back for the trip to Liverpool where he was crucial again in a historic 3-0 victory.
He lost his place again for the next home game, though, as West Ham entertained Newcastle. But when Ogbonna was injured midway through the first half, Tomkins came on in his preferred position to help see out a comfortable 2-0 victory.
If Bilic had not spotted the common theme in West Ham's performances by then, he could not fail to realise it in the next game as Tomkins retained his place at centre back due to Ogbonna's injury and was pivital as the Hammers recorded another away win, this time 2-1 over then runaway leaders Manchester City.
Do England ambitions offer Sunderland hope?
Sunderland might hold out some hope they could land Tomkins if he finds himself on the bench in the near future as Ogbonna returns from injury next weekend.
He has been a first team regular under Allardyce for the last four years and having represented England at every level up to Under 21s would be hopeful of at some stage making senior side.
He won't do that from the West Ham bench, though.
Facts point to future with Hammers
But with the Irons on the brink of their exciting move to the 54,000 seater Olympic Stadium next summer, the facts point to Tomkins staying in East London.
The new five-year deal, the fact Jenkinson is only on loan and will at some stage return to Arsenal where he himself has signed a new long-term contract, his versatility and the fact he is a homegrown Hammer suggest his future remains very much in East London.
We all know money talks in football, though, and is Sunderland stump up enough cash it could change everything.
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.