The 24-year-old could make his debut in Sunday’s Betfred League One derby at Bradford Bulls.
Blagbrough played junior rugby with Newsome Panthers and made one Super League appearance for Huddersfield Giants in 2013.
He joined Sheffield Eagles the following year and also had loan spells with Oldham and York City Knights.
A former University of Leeds student, Blagbrough spent last season at Australian club Mackay Cutters and joined Leigh on a three-year contract in the closed-season.
“He is quality,” Hunslet coach Gary Thornton told the Yorkshire Evening Post.
“He has got quite a number of Championship games under his belt and he had a great season with Mackay Cutters.
“He has got a full-time contract with Leigh, but he is desperate to come and get game time and with our injury situation – Brad Foster, Ryan Mallinder and Nyle Flynn are all out at the moment – he will certainly get that.
“There’s a possibility he could get six games in [during his initial month] and Leigh have said he can play for us in the Challenge Cup.
“It is a massive boost to get a player of his experience and he is coming at a great time for us.”
Hunslet’s director of rugby Darren Williams added: “Jack joins us at a crucial time with a busy period coming up over Easter, with some tough games ahead due to a backlog of postponed fixtures.
“I would like to thank Keiron Cunningham and the Leigh club for facilitating this move so quickly.
“Jack, along with recent loan signing Josh Jordan-Roberts from Leeds Rhinos, will come into contention for this weekend’s game against the Bradford Bulls.”