Coleman guided Wales to an historic run in the European Championships this summer, while Giggs was Louis van Gaal's number two over the past two years
Bob Bradley's reign as Swansea manager spanned just 11 games before being brought to an abrupt halt on Tuesday evening.
His time in charge will go down as one of the shortest in the history of the league, having taken charge for 85 days. Of those games, the American picked up only two wins and two draws leaving the club in a dogfight to retain their top flight status.
The club will now look to rebuild as quickly as possible and reports suggest that two Welsh natives could be targets for the job.
BBC Sport report that Wales international manager Chris Coleman could be approached with an offer to take over the Premier League side. Coleman impressively guided Wales through the qualifying campaign for the European Championships and then helped the country break new ground on the international stage.
Gareth Bale and Hal Robson Kanu starred as Wales made it as far as the semi-final of a major competition for the first time in their history.
They haven't got off to as impressive a start in the World Cup qualifiers, sitting third in Group D behind Serbia and leaders the Republic of Ireland.
The 46-year-old 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 taken charge of English club Fulham.
Chris Coleman led Wales through a remarkable European Championships campaign in France this summer. Image:
The second high profile name in the running for the position is Ryan Giggs, who was twice interviewed for the job before Bob Bradley's appointment.
Giggs served as Louis van Gaal's number two at Manchester United for two seasons and was also interim of the club following David Moyes' departure in 2014.
It is not known whether he has any interest in the job at this point. The Welsh club claimed that Giggs had given an "underwhelming" interview, before the former United midfielder addressed the claim by saying: "I met with them [Swansea] 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."
Other names in the picture are former Derby County manager Nigel Pearson and ex-Birmingham City boss Gary Rowett.