Uruguay international Godin was an integral part of Atletico's breakthrough La Liga title success last season, and their run to the UEFA Champions League final.
The 28-year-old said he was always content to stay at the Vicente Calderon, due to his relationship with coach Diego Simeone.
"From the first moment I got here I was told by the press I had offers," Godin told AS.
"I never knew anything. This year it's true there were several enquiries but let me be very clear: with the confidence Cholo [Simeone] has in me and what he said about me, what the president and the fans have given me and the feeling I have for this club, it's very difficult to leave here.
"It's true that football is a job and you work to make money but when there are so many feelings involved it's very difficult to change your team to earn a little bit more money."
High-profile departures from Atletico in the off-season included striker Diego Costa, and left-back Filipe Luis, but Godin understood their reasons for leaving.
"I respect all players and their decisions but that's how I feel. I decided to stay and hopefully I'll be here for many years," he said.
"The important thing is we kept the core of the team, which gives stability to the club. Now it looks normal but the hardest part is maintaining consistency."
Simeone, whose stint at Atletico began in 2011, has been the cornerstone of the club's revival from ninth in the 2009-10 season to champions in May - according to Godin.
"I had many ups and downs at Atletico, achieving the best and the worst. We won trophies but lost games and missed the Champions League," Godin said.
"With the arrival of Cholo we were stabilised and there have been significant trophies. That gives you everything.
"The line between winning and losing is very thin and wins give you prestige, stability and continuity. And it’s true my career at Atletico has gone from less to more."