The 28-year-old was one of Atletico's standout players last season as he helped the club to win the La Liga title and reach the Champions League final.
"From the moment I arrived here I felt, due to the papers, that there were offers for me, but I never knew anything," Godin told AS. "This year it's true people asked about me, but with the backing of Diego Simeone... and with the feeling I have for this club, it's hard to leave.
"It's true that football is a job and one works to earn money, but when there are so many feelings involved, it's hard to change clubs just to earn a bit more cash.
"I respect every player and their decisions, but that's my feeling. I decided to stay and hopefully I'll be here for many years."
Godin joined Atletico in August 2010 from Villarreal.