The DFB said in a statement on Thursday that Werder had requested the release of Dutt, who succeeded Matthias Sammer at the association after the former Germany defender took up a role at Bayern Munich last year.
Dutt, who joined the DFB in August 2012, has a contract to 2016.
"Because of the importance of the person involved, the DFB board will discuss whether to offer the desired release," the authority said.
The 48-year-old Dutt made a name for himself after guiding Freiburg into the Bundesliga in 2009 and keeping them there until his departure in 2011 for a season at Bayer Leverkusen.
Werder narrowly escaped relegation this season and parted ways with coach of 14 years Thomas Schaaf on the penultimate weekend of the season.
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- Bayern Munich
- Matthias Sammer