Naglesmann took charge of Bayern back in 2021 and guided the club to the Bundesliga title in his first season in charge, as well as the DFL-Supercup in both 2021 and 2022.
Despite this, he was relieved of his duties at the Allianz Arena in March 2023 - which Eberl thinks played a role in his decision to stay with the national team across recent weeks.
Regarding the club looking at other prospects, the Bayern chief added: "Very, very good shape when it comes to searching for a coach and hope that we'll get further there soon.
"I know that time is of course of the essence. That's also our intention, that we want to do it quickly, but we still want to do it as hard and as thoroughly as necessary and that is our job that Christoph [Freund] and I do.
"But as I said, we are now very, very final and will hopefully be soon what we can announce."