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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."

In the other semi-final Salford Red Devils play Wigan Warriors, the only side outside of Super League's top four left in the competition.
Hull were the final side to confirm their place in the final four, after defeating Super League leaders Castleford 32-24 on Sunday.
The semi-finals will take place at neutral venues on the 28 and 29 July.
Hull FC are just one place and one point behind Rhinos in the Super League table, and Hull forward Mark Minichiello told the Rugby League Forum: "It's a tough game. It's at a neutral ground.
"They're playing really well at the moment, they're above us in the table, and they've got their mojo back.
"It's going to be a great outing and I can't wait for it. It's two form teams going at it in the semi-final."
Meanwhile, last season's Super League champions Wigan are down in eighth, while Salford, who only maintained their top-flight status with a Million Pound Game win last season, are second in the table.
Warriors full-back Sam Tomkins told BBC Sport: "I wouldn't say Salford have surprised me this season, they've got talented players.
"With (brother) Logan Tomkins being there, I speak to him about what they do, they've got a good set up.
"It's taken a bit of time for Salford to really get some consistent form and they've got it at the moment."

David McGoldrick volleyed in his first goal since August to put Ipswich ahead.
Revell, who recently saved a penalty as an emergency goalkeeper against Preston, levelled from the spot.
Brett Pitman's glancing header restored Ipswich's lead, and Revell poked home Josh Murphy's free-kick to equalise before Varney's header sealed victory.
Ipswich manager Mick McCarthy: "It was exciting for both sets of fans but frustrating for me.
"We have already started thinking about next season - that is an ongoing process."
MK Dons manager Karl Robinson: "It was disappointing for us but it summed up our whole season. We were good but also poor in certain key areas.
"It has been a massive learning curve. Some of these players deserve to be in the Championship but some don't and that is why we are where we are."

The 23-year-old midfielder has scored 21 goals in 53 league appearances since joining the Gunners from Sunderland Ladies in 2010.
Nobbs has won two league titles, three FA Cups and four Continental Cups during her time in north London.
She has won 24 international caps and helped England finish third at the World Cup in Canada last summer.
Arsenal Ladies have not disclosed the length of Nobbs' new deal.
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.
[NEXT_CONCEPT]
Defending champions Hull FC will play Leeds Rhinos in the semi-finals of the Challenge Cup.
[NEXT_CONCEPT]
Luke Varney's late winner secured Ipswich's first win in eight games, despite Alex Revell's brace for already-relegated MK Dons.
[NEXT_CONCEPT]
England international Jordan Nobbs has signed a new contract with FA Women's Super League 1 side Arsenal Ladies.