The 30-year-old was heavily linked with both Arsenal and West Ham over the summer, with his time in Turin looking to be set to end sooner rather than later.
Quagliarella is believed not to be seen as a key first-team player by Juve boss Antonio Conte, who has handed him just three Serie A starts this season.
A winter switch away from the Italian champions looks to be on the cards, and both Southampton and Swansea are keen to offer the 30-year-old an escape route.
Having represented his nation on 25 occasions – including at the 2010 World Cup – the former Napoli ace’s experience and top level know how is thought o appeal to both sides.
Swans chief Michael Laudrup faces an arduous latter half of the season, with his side having advanced to the knockout phase of the Europa League.
Fresh faces are needed to boost his side’s chances of impressing both continentally and domestically, with Quagliarella seen as a wise addition.
Southampton are also watching the forward, with fresh faces wanted by Mauricio Pochettino – who is looking to build on his team’s impressive start to the season.