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Ryan Giggs was the favourite following Bob Bradley's exit on Tuesday, but the club are now looking elsewhere.
Wales boss Chris Coleman was also a target but is out of the running.
Clement was interviewed for the job in October before Bradley replaced former head coach Francesco Guidolin.
The former Derby County manager, who was sacked by the Rams in February, has spoken to the Swansea hierarchy having impressed them in the autumn.
Rowett, meanwhile, has not taken on a job since his sacking by Birmingham City earlier in December.
The Swans are hopeful of having Bradley's replacement in place before the trip to Crystal Palace on 3 January.
Alan Curtis, who is taking temporary charge of the team, believes the club's next manager should be British and have experience of the Premier League.
"It has to be the right man. Ideally they would be British and know the Premier League," Curtis said.
"[The owners] have to be satisfied he fills all the criteria for getting us out of the trouble we find ourselves in."
Former Swansea forward Curtis, who is preparing for Saturday's home game with Bournemouth, says he has spoken to chairman Huw Jenkins but does not know how long he will be in charge.
"We will take it on a day by day basis," he continued.
"What the club needs is not a change of direction. The club has been the club and the way we run the club is the way that suits us.
"Somebody needs to come in with our philosophy with a little bit of experience and perhaps look at it with fresh eyes. That's exactly what it needs."
Bayern Munich assistant manager Paul Clement is the leading candidate to become Swansea City's new manager - but former Birmingham City boss Gary Rowett remains in contention.