Billy Davies is the overwhelming favourite for the post after Blackpool confirmed the Scot had "decided to follow another opportunity" rather than consider their approach.
In the meantime, McLeish's departure after just 40 days has left assistant boss Kelly - a former Leicester manager - to take the team to Bristol City at the weekend.
Chairman Fawaz Al Hasawi said: "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."
Forest have also confirmed that two of McLeish's backroom staff - coach Peter Grant and Dr Ian McGuinness - have left the club.