Bradley was sacked on Tuesday after just 85 days in charge, with the Swans 19th in the Premier League.
Giggs, 43, was interviewed twice before Bradley's appointment in October.
Wales manager Chris Coleman, former Leicester City boss Nigel Pearson and ex-Birmingham City manager Gary Rowett are also in the frame.
Bradley's permanent successor is not expected to take charge before their Premier League match at home to Bournemouth on New Year's Eve.
First-team coaches Alan Curtis and Paul Williams will take the reins temporarily for that game, but the Swans are keen to make a swift appointment.
Boxing Day's 4-1 home defeat by West Ham was Swansea' seventh in 11 games since Bradley took over and leaves them on 12 points, above bottom side Hull on goal difference.
At the time of Bradley's arrival, Swansea were above the relegation zone on goal difference but have picked up just eight points out of a possible 33 since, conceding 29 goals in the process.
'Giggs and Swansea a good fit'
Giggs' only managerial experience was a four-game stint as Manchester United's interim player-manager at the end of the 2013-14 season.
He had worked under David Moyes earlier in that campaign and was assistant manager to Louis van Gaal afterwards, but left in July following Jose Mourinho's appointment.
The former Wales captain spoke with Swansea's hierarchy on two occasions before Bradley was appointed, and the club's chairman Huw Jenkins is thought to have been impressed by Giggs.
Rene Meulensteen, Manchester United's former first team coach, believes Giggs would be well suited to Swansea.
"I don't agree with Ryan not having management [experience]. He's got his apprenticeship in his time as an assistant manager at Manchester United," the Dutchman told BBC Radio 5 Live.
"Is it the right time for him? He has to start somewhere. And in many ways when he was linked with Swansea [in October], I thought it would be a really good club for him.
"The club needs to sort itself out now because it's lost its stability and continuity it's had over the last five or six years."
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