The midfielder will sign a two-year deal with the Brewers when his Fratton Park contract comes to an end at the end of June.
The 25-year-old had been told he was free to join a new club by Danny Cowley following the conclusion of the season.
He spent the second half of the 2020-21 campaign at the Blues’ League One rivals Northampton but was unable to help them avoid relegation.
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Morris played 22 times for the Sixfields outfit, after featuring 18 times for Pompey over the first half of the season.
The former England youth international made 28 appearances for the Blues after arriving from Shrewsbury in January 2019.
Yet his time at Fratton Park was blighted by an abdominal problem which ruled him out for 14 months.
Morris spent time on loan with Burton during boss Jimmy Floyd Hasselbaink's first spell in charge in 2014.
He said playing for Hasselbaink was again was a big lure.
‘It's good to be back. I was only here for a month, but I really enjoyed it,’ Morris said.
‘The manager got in touch and that was a big point for me - it showed he was keen to get me.’
Morris joins Jack Whatmough (Wigan), Ben Close (Doncaster Rovers), Ryan Williams (Oxford) and Tom Naylor (Wigan) in finding new clubs following their Fratton Park release.
All five are due to face the Blues next season in League One.