Gartside, a member of the Football Association council, added: "I am flattered Sam's name has been linked with the England coaching vacancy.
"Until such time, Sam is 100% committed to his important role at Bolton.
"He is excited about the prospect of guiding the club on three major fronts - the Uefa Cup, the FA Cup and the possibility of yet another top-half finish in the Premiership."
Gartside also insisted Allardyce isn't interested in applying for the Newcastle United job.
He said: "I have asked Sam about the speculation linking him with other Premiership managerial vacancies.
"He has made it perfectly clear to me he is not interested in managing a Premiership club at any of the current vacancies."
Sam Allardyce himself admits he's ready to be interviewed for the England job.
He told Sky Sports News: "I feel I have done my apprenticeship and, at this stage of my career, I would have to say I would not be frightened of the job.
"It is flattering to be talked about for the job and I hope now it goes to the next level.
"I would love the opportunity to be interviewed for it."