The former midfielder, who was part of the Arsenal team that last claimed the Premier League title in the 2003/04 season, took on the role of the club's inaugural technical director in 2019.
At 46 years old, he played a pivotal role in the selection of Mikel Arteta as manager later that same year.
Reports indicate that the Brazilian may be considering a position with Greek businessman Evangelos Marinakis, who owns both Nottingham Forest and Olympiakos.
"This was an incredibly difficult decision to make," Edu stated in a release on Arsenal's official website.