That is according to club legend Martin Keown, who admits he was surprised by the news.
Edu has been a key reason for the Gunners’ revival in recent seasons, as they finished second in the last two Premier League campaigns.
He is now being linked with a role at Evangelos Marinakis' group of clubs - Nottingham Forest, Olympiacos and Rio Ave.
"Well, I don't know for me what could be better than being in that role for Arsenal," Keown said on talkSPORT.
"He may want to return home to Brazil."
Keown added: "Ultimately, it took him a while to get going, didn't it, Edu? I mean, he needed some support from those above him.
"I think the biggest change in Arsenal was a greater investment in time and inclination from the Kroenke family. And I think that's been mirrored in Josh Kroenke becoming much more visible and much more engaged with Arsenal post the challenges of the European Super League and Arsenal's climb down from that position.
"So I think ultimately, if they've got a good relationship, good relationships are nice to have. But Arteta, does he lose a big ally? Does he need a big ally?
"Does he already have a relationship with those that make the decisions? I would suspect so. I would suspect that ultimately. Will it be an advantage? It depends if they bring a better sporting director in.
"If they bring a better sporting director in, that's got a wider brief, that's got a better understanding of being able to acquire even better players than the ones that Edu did... I'm not entirely sure that all of Edu's signings were the greatest signings.
"So with that in mind, you've got Arsenal being challenged on the field right now and people perceive it as being challenged off it now."