Wales boss Coleman is on Swansea’s shortlist along with Ryan Giggs, Gary Rowett and former England boss Roy Hodgson to replace Bob Bradley .
But Swansea believe they can prise Coleman away despite his remarkable success with Wales after leading them to the semi finals of Euro 2016.
Coleman, 46, is on £500,000-a-year and under contract until after the 2018 World Cup but Swansea could offer a bigger deal which could tempt him.
Welsh Football Association bosses fought off interest from Hull at the start of the season but are braced for another approach from Swansea.
Swansea chairman Huw Jenkins is known to be an admirer of ex-Manchester United star Giggs but that would be seen as a major risk as it would be his first big job.
Former Birmingham boss Rowett is on Swansea’s radar as they look to make a young British appointment but no approach has been made.
Hodgson would also be keen on the job and has made it clear he wants to get into management while Nigel Pearson has a track record after guiding Leicester to safety and even Harry Redknapp has thrown his hat in the ring.
Swansea pulled the trigger on Bradley without having a definite man in mind and will now interview the contenders before trying to finalise the appointment before next Tuesday’s trip to Crystal Palace.
Alan Curtis and Paul Williams are expected to take Saturday’s game with Bournemouth.
Swansea did not want to sack Bradley but viewed the defeat to West Ham as a “game changer” and feared the former US boss could not turn things around.
Bradley, sacked just three months into a three year contract, said yesterday on Wednesday: “‘I’m a little bit p***** off this morning. I don’t think it’s the correct decision.
Bob Bradley’s last interview as Swansea boss
“I believe in my work and I certainly knew that I was going into a difficult situation and I also understand that when you go in the clock’s already ticking, so it’s not like you’re expecting all sorts of time.
“The discussions we had always included the work that needed to be done in January – we had talked about players. I’m frustrated because I feel like every place I’ve been, I’ve been able to put my stamp on the team in terms of the mentality and the tactics.”