The Daily Star says Toon have tabled a £4m bid for Tomkins.
And their hopes of landing the defender received a huge boost yesterday when he told West Ham he wants to quit.
Toon boss Alan Pardew has had two bids for Tomkins - including a second of £2.5m - rejected.
But Newcastle upped the stakes to £4m yesterday and Pardew hopes it will be third time lucky.
The Geordies raised the bar after learning of rival interest from Tottenham.
Tomkins, who is now looked after by boss Sam Allardyce's representative Mark Curtis, is understood to have handed in a transfer request yesterday.