The 25-year-old struggled to make an impact for the Blades in the first half of 2021/22 and spent the second half on loan at The Den.
Injuries hampered his effectiveness but Burke did manage 14 appearances for the Championship club, scoring twice and providing one assist, and certainly seems to have won over the manager.
The Scottish attacker is now back with parent club United, where he is under contract until next summer, and Rowett has revealed their stance on him will dictate Millwall’s approach.
Speaking to News At Den about the prospect of re-signing the 2022 January arrival, the Lions boss said: “Oli had a good time here last season. If Sheffield United decide that he’s not part of their plans, then of course he’s the type of player that we’d like to bring back in.
“It’s the level of player that we’d like to try and bring in this summer, and we’ll work incredibly hard to make that happen.
“It’s easy to look and say that we haven’t done much yet, but there’s been a lot of work behind the scenes that I believe will come to fruition in the next week.”
Stoke City striker Benik Afobe and Arsenal defender Daniel Ballard also impressed while on loan at Millwall last season.
Quiz: Can you name which club these 27 ex-Millwall players are playing for now?
Rowett clearly feels he can be the one to unlock the potential that Burke has.
The 25-year-old is not short of raw talent but has struggled to find any sort of consistency in recent years.
He did look fantastic in glimpses last term and the Lions boss may feel a cheap deal is there to be done if the Blades want to offload the Scot.
With Jed Wallace’s future up in the air, attacking reinforcements are necessary but Burke still represents something of a risk.