Gunnarsson was a star of Euro 2016 earlier this summer, captaining minnows Iceland to the quarter-final stage after an unbeaten group-stage campaign and a famous win over England in the last 16.
The 27-year-old has made 197 appearances for the Bluebirds across five seasons in the Welsh capital, but fell out of favour during the latter weeks of last term under Russell Slade and was left out of the season opener at the weekend by new manager Paul Trollope.
That has led to speculation that his future could lie away from Cardiff, and according to the Sun, both Derby and relegated Norwich are keen on adding the combative leader to their ranks.
The paper suggests that a fee of £2 million would be enough to secure the Icelander's services, which could be a bargain considering what he can bring to a side.
Gunnarsson has 330 appearances in the Championship and the Premier League to his name with Coventry City and Cardiff, and has also clocked up 63 caps for his country, whom he has captained for the last four years since the age of 23.
Though he signed a new three-year deal last summer, the Welsh club are only looking to recoup a small fee for the man they signed on a free transfer in 2011.
Both Derby and Norwich are thought to be on the lookout for more players to add to their squads, and Gunnarsson would fit the bill nicely in the centre of midfield.