HITC Sport understands that the Spaniard has emerged as a late window target for Canaries boss Alex Neil.
The Canaries, who have collected four points from their opening three games back in the Premier League, are aware that they are running out of time in their efforts to get further fresh faces on board before the close of the transfer window.
Their efforts will be stepped up before Tuesday’s deadline, and Amat has been identified as a possible addition.
He has spent the last two seasons with Swansea, having been snapped up from Espanyol in 2013.
The Spanish centre-half has been unable to hold down a regular place in the Swans side, partly due to untimely injuries, with Ashley Williams and Federico Fernandez currently ahead of him in Garry Monk’s plans.
Amat is, however, held in high regard at the Liberty Stadium and penned fresh terms through to 2018 in March – as reported by the BBC.
Norwich, though, are prepared to test Swansea’s resolve when it comes to their desire to keep the former Spain U21 international in South Wales.