Former chairman Peter Hill-Wood was forced to stand down from the role because of health reasons, and the Gunners have turned to British businessman Keswick to fill the vacancy.
But Adams said in The Sun: "Chips is a great guy but not a very imaginative choice by the owner - and he is 73.
"If they just wanted a figurehead, they should have gone for me. It would have been a better visionary decision than Chips.
"They should be lining up the new manager. Chief executive Ivan Gazidis is in a tricky situation and could do with some help.
"It's time that Arsenal won something again, even the FA Cup or League Cup. But I can't see it, to be honest. They are still not good enough in certain areas of the team. And they are miles off the title."
Adams made 669 appearances for the North London club and won 10 major trophies.