The 17-year-old was the third player to be taken in under the club's apprenticeship scheme launched by former coach Paul Thompson and is currently in the second year of the programme.
A product of the Bradford Bulldogs youth system, Brown has made 10 appearances so far for the Steelers this season.
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Injuries meant he played a more significant role in the 5-2 defeat at Nottingham last Saturday, but he was a mere bystander in Wednesday night's 3-1 defeat to Fife Flyers at Sheffield Arena, standing at the back of the bench throughout the game alongside fellow apprentice, defenceman Jordan Griffin.
In a press release from the Hawks, Steelers' owner Tony Smith said: "Kieran is a fantastic young talent but like all young players, just wants to be on the ice every weekend.
"He thoroughly enjoyed his experience playing at Blackburn last season, joining up with players he came through the youth system at Bradford with. The deal will be beneficial for both Steelers and Hawks as he gets more ice time and continues to develop."
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Hawks player- coach Steve Duncombe - a Sheffield-born defenceman who has played the majiority of his career with the Sheffield Steeldogs - is hoping Brown's registration can be put through in time for him to play in Saturday night's NIHL North Cup encounter at Jason Hewitt's Hull Pirates.
Griffin, taken in under the apprenticeship scheme this summer after also coming through the Bradford junior programme, is currently on a two-way deal with Sutton Sting, weho also play in the NIHL One North.
READ/WATCH MORE - Match report: Sheffield Steelers 1 Fife Flyers 3