The veteran departed Atletico at the end of June following the expiration of his contract and was strongly linked with a return to former club Chelsea.
However, in a dramatic turn of events, Tiago signed a new deal with Atletico.
"At the end of the season I received a very good deal, an offer that was important not only as a player but also for the future, and it made me think about leaving," he told Marca.
"There came a time in the summer when I realised that the most important thing for me was to feel important at the club. I had options to go, better options in terms of money. I'm not here for money but for the feeling of these colours.
"I spoke with Cholo (Diego Simeone) and he told me to stay for the next training session. At that time I only thought about returning, wearing this shirt again. I'm happy it's finished. I'm very happy, grateful to the club, players, coaching staff and the directors for having me back in the team."
The 33-year-old has also played for the likes of Benfica, Lyon and Juventus during a well-travelled career.