Both Stewart - the former Hull City winger - and defender Brown are free agents and have been on trial at Bramall Lane.
Brown will sit down with the Blades today to thrash out a permanent deal, while Stewart has also impressed boss Chris Wilder.
With the emergency loan window scrapped, United have looked to free agents to provide extra cover at full-back and wide midfield.
“We’ll be speaking with Reece today,” said Wilder. “He’s done well, impressed us, and I’ve spoken before about the importance of having competition for places through the team.
“I’m not going to be sticking a gun to anybody’s head. But, like I say, he’s done well and so we’ll sit down and talk about something.
“Cameron is still here with us and he’ll play in a game today. We’ll probably reach a decision on him at some point after that. We’ll see what happens there.
“Players can find themselves without a club for all sorts of reasons. They’ve both been given opportunities here and it’s a chance for one, potentially for both, to put their careers back on track.
“We tried to get another wide player in before the window closed but we weren’t able to get anything across the line in time. That’s why we are taking a look at him, because that vacancy is still there.”
Stewart has had loan spells at Doncaster Rovers, Barnsley and Leeds United - having started out at Manchester United - but the 25-year-old without a club.
Brown, 24, has played for Barnsley and Bury in the last two seasons.
The Blades are hopeful that Caolan Lavery and Leon Clarke will be fit for Saturday’s trip to League One leaders Scunthorpe United.
Lavery has a sore ankle, but trained today, while Clarke has had a scan on his troublesome foot/ankle.
Wilder confirmed: “Caolan will train with us today and so he should be in the squad. He took a bit of a whack to his Achilles against QPR and so we brought him straight off.
“We didn’t take any risks. But he’ll be involved and we’re hoping he’ll be fine. Obviously it was a bit sore for a day or so but you expect that to be the case.
“Leon is going for a scan and we’ll get the results back from that later. Hopefully, when we do, he’ll be okay for the game as well.
“He’s carried it for a while and obviously 19,000 or so people saw him wearing a protective boot last weekend. The scan is just so we can get to the bottom of what it actually is.”