The 29-year-old, who has featured three times for the Foxes this season, moves to Ashton Gate for a period of one month.
Robins manager Derek McInnes said: "I'm very pleased to be able to bring Neil to the club. He's hungry and determined to do well. He's a tenacious midfielder who brings experience and quality to our squad.
"He hasn't played as many times as he would have liked for Leicester for this season, but given the amount of quality players they have and the form they are in, that's nothing to be ashamed of.
"Neil is desperate to play games for us, he brings another level of quality to our squad he can't wait to get going."
Danns is expected to make his debut in Saturday's home match against Blackpool.