The Eagles are understood to be keen on the six-cap England international, who has shone since leaving Liverpool to join the Swans in 2013.
However, reports claim that Newcastle are also interested in the 23-year-old, as manager Steve McClaren looks to inject some energy and tenacity into his relegation-threatened side’s midfield.
The Magpies are currently stuck in the relegation zone – coincidentally two points behind 17th-placed Swansea – after a difficult start to the season.
Shelvey is understood to have grown unsettled at the Liberty Stadium following the recent sacking of Garry Monk and was left out of stand-in manager Alan Curtis’ squad for the weekend loss at Manchester United.
Interestingly, Palace manager Alan Pardew worked with the dogged central star during his spell at Charlton Athletic, which could be a factor in the player’s eventual decision.