The Blues boss is keen to bolster his squad with the former Rotherham winger, who is a free agent after being released by Sky Bet Championship newcomers Doncaster at the end of last season.
Woods has been training with the Spireites and recently featured in a couple of behind-closed-doors matches, impressing Cook on both occasions.
After some high profile signings this summer, Chesterfield are currently over budget and may need to move someone out in order to accommodate the 27-year-old.
A board meeting took place on Thursday and Cook hopes chief executive Chris Turner and chairman Dave Allen might make some funds available to bring in Woods.
"I know there's been a lot of talk about budgets, but we are always looking to improve," said Cook. "It's an ongoing thing, we are always interested in good players.
"Martin Woods has been with us. Players of the quality of Martin Woods are never on trial at a League Two club so I won't do him the disrespect of saying he is on trial.
"He has played for us in both of our behind-closed-doors matches in the last week or so at home to Grimsby and at Sheffield Wednesday on Tuesday and he played very, very well. He is a quality player."
Asked if he would like to add Woods to his squad, Cook replied: "We are always interested in good players and we are always looking to improve what we have, and Martin Woods is a very good player.
"There's a board meeting this afternoon so I'll try my arm again. It's my job to pester the board."
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- Martin Woods