Godin forced off at half-time of Atletico's 2-0 home win against Getafe on Saturday.
"Our captain has undergone an MRI at the Clinica Universidad de Navarra and the results offered by the medical services of the club indicates that he suffers an elongation in the adductor muscle of the right thigh and remains pending evolution," a club statement said.
Godin, who is out of contract at the end of the season, has been heavily linked with a free transfer move to Inter in recent weeks.
Atletico coach Diego Simeone admitted earlier this month that Godin could leave in the summer, though he denied suggestions that the club are making no effort to keep the defender.
In the short-term though, Godin joins a concerning injury list for Simeone with defenders Stefan Savic and Filipe Luis already sidelined along with Spain internationals Koke and Diego Costa.
They have, however, bolstered their forward line after agreeing a loan deal with Chelsea for Alvaro Morata.
The former Real Madrid forward has signed an 18-month loan deal with Atletico holding the option to make the move permanent at the end of that time.
Morata is in line to make his debut on Sunday when Atletico travel to Real Betis.
They then host Real Madrid the following week before taking on another local rival in Rayo Vallecano on February 16, four days before that huge European tie against Italian champions Juventus.
Simeone's side are second in La Liga, five points behind leaders Barcelona, following Sunday’s win at the Wanda Metropolitano.
Antoine Griezmann and Saul Niguez scored in the first half to seal victory for Simeone’s side, with Getafe’s Djene Dakonam and Leandro Cabrera sent off late on - both for two bookable offences.
Atletico have now won six of their last seven games in the league and have lost just once in La Liga all season.