Vale have targeted the 39-year-old former Newcastle and Portsmouth defender after being hugely impressed by his work in non-league.
Stimson took Grays Athletic to the Blue Square Premier - formerly the Conference - in 2005 and has won the FA Trophy for the last three years in succession with Grays and Stevenage.
But Stimson is under contract with Stevenage until 2010, so Vale face a compensation claim if they press on with their interest.
Gillingham have also been denied permission to speak to the manager by Stevenage chairman Phil Wallace.
Stimson said today: "I'm flattered by the interest and it's an ambition of mine to manage in the Football League. Phil Wallace refused me permission and that's that."
Stimson's name is likely to remain high on the list of possibles when the Vale board meet tonight to discuss their shortlist. Caretaker manager Dean Glover wants the job, while ex-Aston Villa and Stoke boss Brian Little has also been linked with the vacancy.
Stimson has guided Stevenage to second in the Blue Square Premier this season, following their FA Trophy victory last term.
Vale chairman Bill Bratt has revealed he has received a sizeable amount of interest in the manager's job, but it remains to be seen how determined he is to get Stimson to the club.
Source: http://www.thisisstaffordshire.co.uk/