The Gunners missed out on their preferred option to take over as technical director at the Emirates when Monchi returned to Sevilla on Sunday, having left Roma recently.
Reports suggested that Overmars was next in line for the job, but he penned a new deal with Champions League quarter-finalists on Monday.
"There are still plenty of challenges at Ajax and I'm having a good time here,” he said.
“We are taking steps with both the youth academy and the first team. From a sporting point of view, we still want to show a lot.
“What is also important to me is that it's nice to work with the people around me. I get pleasure and satisfaction from that. "
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Overmars spent three years as a player at Arsenal and won the FA Cup and Premier League while in London.
He joined the Ajax backroom team in 2012 after a spell with Go Ahead Eagles, and the club have since gone on to win two Eredivisie titles and reach the Europa League final in 2017, where they were beaten by Manchester United.
This season the Dutch side stunned Real Madrid with a 4-1 Champions League last-16 second-leg win at the Bernabeu which sent them through to the quarter-finals with a 5-3 aggregate victory, where they will face Juventus.
