The Scotland international rose to prominence in 2016 when, on the back of the promise he showed as a teenager at first club Nottingham Forest, he moved to RB Leipzig for what was believed to be a record fee for a Scottish player.
Now the 25-year-old is heading back to the Bundesliga after a spell at Bramall Lane which, like his time in Leipzig, failed to hit the heights.
As is increasingly the case these days, the transfer fee is undisclosed.
Burke scored just two goals in two seasons with the Blades, the first of them ending in relegation from the Premier League. He did, though, score the winner in a 2-1 victory at Manchester United.
Generally, though, he struggled to make an impact and was loaned to Millwall for the second half of last season, scoring twice for the Lions.
It had been thought a permanent move to South London might be possible, but Bremen have got there first.
Die Grün-Weißen dropped out of the German top-flight in 2021, but won an instant return with last season's second-place finish.
Although Burke started well in Leipzig, with an assist and a goal in his first fortnight, he was limited to a largely bit-part role and is still to start 20 league games for any club. After coming back from Germany he played at West Bromwich Albion, and on loan with Celtic and Spanish side Alaves. Now 25, he really needs to establish himself.
With David McGoldrick and Lys Mousset released at the end of last season and Oli McBurnie without a league goal since July 2020, the Blades need striking reinforcements as they look to improve on last season's Championship play-off semi-final.
They are yet to make a signing in any position this summer.