The England U-21 international has not featured for United's first team this season, although he made eight appearances during a month-long loan spell with Derby before returning to Old Trafford in January.
Keane, who spent the majority of last season on loan at Leicester, had been widely linked with a move to Rovers in recent weeks.
The 21-year-old has now completed his temporary switch, and he has joined in time to be considered for his Rovers debut in Sunday's derby against Burnley.
"I'm delighted to finally join," said Keane. "It has been a rumour for a while now, so I'm just glad to get it all sorted.
"I've got a bit of experience now in the Championship and I know what it takes to play well in these games, so I'm looking to play well on Sunday and hit the ground running and keep that form up.
"I'm sure it will be a tough game, but I'm confident we can get a result. I'm really looking forward to it. I hope it is my debut and I'll make sure I'm ready to do well."
Rovers boss Gary Bowyer was delighted to bolster his injury-hit squad with the signing of Keane, who he described as "highly sought-after".
"He has had a good education with Manchester United and he has got experience already in the Championship with his time on loan at Leicester and Derby - two top clubs," Bowyer said.
"We are delighted to have been able to secure his services until the end of the season."