The Royals have tabled a £7m offer for the wide man, whilst the Merseyside club have seen a previous £4.5m bid rejected.
Reading looked to be in the clear to pursue their target, with Liverpool having snapped up Philippe Coutinho from Inter Milan, but it now appears that Swansea are seriously interested.
The League Cup finalists are believed to be in the market for attacking additions, with manager Michael Laudrup keen to build on an impressive start to the season.
A creative wide midfielder is thought to be of interest to him, with club having sold Scott Sinclair to Manchester City over the summer.
Reading are desperate to land the skilful winger and are willing to splash the cash to do so.
Despite some impressive recent results, the Madejski Staium-based side are sill stuck in a relegation dogfight.
Manager Brian McDermott sees Ince as the ideal man to add some flair to his attack and offer support to both Pavel Pogrebnyak and Adam Le Fondre.
Liverpool are still thought to be monitoring the situation and could lure the player back to Anfield.
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