The Spaniard, who began his managerial career with the Welsh outfit in 2007, impressed during his first season in charge at Everton by guiding them to fifth in the Premier League table.
Before his coaching career, Martinez spent three years with Swansea as a player, where he helped the club beat relegation to the Conference in 2003.
According to BBC Sport, the 40-year-old has refused to rule out a return to Swansea, while adding: "Swansea City will always be a special memory and a special football club to always keep an eye on.
"Being able to work with [chairman] Huw Jenkins and the board - they are the ones that kept that philosophy and a real clear way of playing the game, which has always been admired around the football world. The achievements have been terrific. As a football club it is very close to my heart and that will never change."
Swansea are currently under the leadership of former captain Garry Monk.