Kelly, who was assistant manager to Alex McLeish before his departure on Tuesday, accepted the caretaker role after a meeting with owner Fawaz Al Hasawi.
The 48-year-old will take care of first-team affairs until the club appoint a new manager. Former Forest boss Billy Davies is favourite to return to the City Ground after rejecting an offer from Blackpool.
Al Hasawi told the club's official website: "I've spoken to Rob and he is happy to take charge of the team for the foreseeable future, starting with Saturday's game at Bristol City.
"He's an experienced man and I know the team is in safe hands."
Meanwhile, the club also confirmed Peter Grant and Dr Ian McGuinness, who were appointed to the coaching staff by Alex McLeish, have left their roles.