The Swans are keen to bolster their options ahead of a debut Premier League campaign and are looking to beat off interest from rivals Cardiff and QPR.
A Swansea spokesman confirmed the approach, telling the Newcastle Chronicle: 'We have expressed an interest in Routledge and are in discussions with Newcastle.'
Confirmed interest: Swansea City have made an offer for Wayne Routledge (right)
Routledge spent the second half of last season on loan at QPR after falling out of favour under Alan Pardew.
He was a key player in the Hoops' Championship-winning side but the Loftus Road club are stalling in the transfer market.
Bitter rivals Cardiff are also in interested in Routledge as they look to rebuild under new boss Malky Mackay.
Out of favour: Routledge rarely featured for Newcastle last season
Routledge had a spell on loan at the Bluebirds in 2008 but it is though he would prefer a move to the Liberty Stadium.
And Swansea are looking to steal a march on their rivals by snaring the former Crystal Palace and Tottenham player.
Routledge would become Swansea's four signing were he to join. Striker Danny Graham, defender Steven Caulker and goalkeeper Jose Moreira have already signed.
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