Eberl, who had spent 15 years at Bayern Munich as a youth player and a pro until 1994, had previously served as Borussia Monchengladbach sports director for 14 years before joining RB Leipzig in 2022.
Leipzig sacked him in September after media speculation linking him to a move to Bayern.
"It is a great joy to come back after 30 years, the joy of returning home," Eberl, who will become board member for sport, told a press conference in Munich.
"The goal is in these three months to get the most out of the season. It is unusual to be in the chasing role eight points behind in the league."
"We are also on the back foot in the Champions League but that also is not impossible to turn around. Maybe even win a title. You never know."
The Bavarians, who last week decided to part ways with coach Thomas Tuchel (50) at the end of the season following a bad run, are eight points behind leaders Bayer Leverkusen in the league.
They also lost their Champions League Round of 16 first leg 1-0 to Italy's Lazio. The return leg is in Munich next month.
"It is a challenge but I don't want to write off the season just yet. There are still too many points left. Now it is about not gifting away the next three months."
Last week Bayern ended a three-game losing run across all competitions with a 2-1 home win over RB Leipzig in the league.
"I saw the team's reaction last week. I have a good feeling. Me and us, we will do everything to realise the impossible," Eberl said.
He said once he starts on March 1st he will also start working with club bosses on finding a successor for Tuchel for next season.
Tuchel will end his spell after just one full season in charge, having taken over in March last year, and with a record-extending 12th straight league title looking doubtful at the moment.
"There is a lot of speculation at the moment (on the next coach) but these are talks and thoughts that we need to make calmly. We have to find a fitting coach for Bayern and then the fitting players for our club and our coach," Eberl.
"For us it is about finding the right coach for Bayern."
