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Bob 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.
Former Leicester City manager Nigel Pearson, ex-Crystal Palace boss Alan Pardew 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's 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.
Swansea Chairman Huw Jenkins is known to be a huge admirer of ex-Wales winger Giggs, who has taken some time out after leaving Old Trafford in the summer following two years working as Louis van Gaal's assistant.
However, after being snubbed in favour of Bradley, it is not known whether Giggs retains an interest in the job.
Swansea-born Coleman, 46, spent four years with the club as a player at the start of his career and has experience in the Premier League as a manager, having spent four years at Fulham after taking over from Jean Tigana in 2003.
He signed a two-year contract extension with Wales in May before guiding them to the semi-finals of Euro 2016 this summer.
However, it is understood Swansea could offer him a bigger salary than the one he is currently on and with Wales presently third in their qualifying group, there is a possibility they may miss out on a place at the 2018 World Cup.
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 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."
Swansea City hope to have a new manager in place by Monday, with Manchester United legend Ryan Giggs and Wales boss Chris Coleman the leading candidates.