The 18-year-old, who made his Premier League debut for the champions in December during a 1-0 win over Reading, has been registered in time to face Arsenal in the FA Cup this weekend.
Rovers boss Michael Appleton is delighted with the signing of a player that he worked with at Portsmouth last season, where Rekik made eight appearances during a loan spell.
Appleton told the club's official website: "He's 18 years of age, a Dutch youth international and has great experience in terms of the football club he's playing at in Manchester City.
"He played against Reading earlier on in the season, he's been on the bench in the last couple of months, he's a player who can play left-sided centre half or left-back, so he can provide us sufficient cover in those areas.
"The boy is full of quality. I'll put my neck on the line and say that the boy will be a Premier League player for many years.
"City are very lucky and I feel lucky that we are able to bring him in for the short-term.
"Hopefully we can use him to the full extent. I'm sure he'll be a great asset and hopefully push us towards what hopefully would be a great end to the season."
Rekik joined Roberto Mancini's side in 2011 after coming through Feyenoord's youth academy and signed a long-term deal at the Etihad Stadium earlier this season.