Briscoe, 22, made three Super League appearances for Leeds in 2014, as well as two on loan at Wakefield Trinity Wildcats.
He also featured for Hunslet on dual-registration two years ago, before signing for Hawks last season on a one-year deal.
He played 17 times for the south Leeds club, but his season was ended by a serious knee injury and was a free-agent before Rhinos re-signed him.
Briscoe, younger brother of Leeds’ currently injured winger Tom, could go straight into the team for Friday’s crucial Super League game at Huddersfield Giants.
Also drafted into the squad is 17-year-old academy loose-forward Cameron Smith, who could make his senior debut.
Zak Hardaker (concussion) and Brett Delaney (hamstring) were both named in Rhinos’ initial 19-man squad on Wednesday, but coach Brian McDermott confirmed they will not feature after being injured in last week’s defeat at St Helens.
Danny McGuire, who has missed the last two games due to a calf problem, is in a similar situation.
McGuire and Hardaker have been removed from Rhinos’ squad for Friday, though Delaney remains to make the list up to 19 names.
Tom Briscoe, Joel Moon, Ryan Hall, Beau Falloon, Adam Cuthbertson, Carl Ablett and Stevie Ward are also ruled out.
Rhinos chief executive Gary Hetherington said: “We have looked after all Luke’s surgery, recovery and re-hab, to such an extent he is now able to play.
“We have found ourselves in this position and he fits the bill, so he has got an opportunity. It is opportune for both parties and we have signed him until the end of the year.”
Brett Ferres (calf) and Josh Walters (foot) could return after injury and Jordan Baldwinson, who played on dual-registration for Featherstone Rovers last week, is also in contention.