Bradley was fired on Tuesday, after only 11 games in charge of the Welsh club.
“Unfortunately things haven’t worked out as planned and we felt we had to make the change with half the Premier League season remaining,” chairman Huw Jenkins said in the statement confirming the club’s decision.
“With the club going through such a tough time, we have to try and find the answers to get ourselves out of trouble.”
Giggs is the 4/6 favourite to take the vacant position, ahead of Gary Rowett, Chris Coleman and Alan Pardew.
The former Wales international almost took over at Swansea, but he was not appointed at the time due to “mixed messages”.
“There I was enjoying my year out, and I get the call off Swansea,” Giggs told ITV.
“I met with them a couple of times but in the end there was just mixed messages from the football side and the ownership side of the club.
“I just felt their ambitions did not really match mine, so it didn’t quite work out.”