The 34-year-old's current deal runs out at the end of the season.
Toure looked like he would be leaving the club after being left out of City's Champions League squad and did not play a Premier League game until November.
But he fought his way back into Pep Guardiola's plans and Saturday's win against Leicester was his 29th appearance of the season.
Meanwhile, Guardiola has called Pablo Zabaleta a "real legend" of the club after the Argentina full-back announced on Saturday that he will leave at the end of the season.
Unlike Zabaleta, who leaves City after nine seasons, Toure is keen to stay on, with Guardiola believing the Ivorian is worth an extra season.
City will honour the departing Zabaleta at Tuesday's final home game of the season against West Brom.
"Pablo is a real legend in what he has done in this club," said Guardiola.
"He helped us with his team-mates, with the club, to make a step forward, to try to qualify for the Champions League, to win the Premier League, to be in Europe three or four times in a row when it was never in Europe for 30 years.
"That's why Pablo Zabaleta is so, so important."
Zabaleta, who signed from Espanyol in August 2008, is City's third longest-serving player after Joe Hart and Vincent Kompany.
He has made more than 322 appearances for the club, winning two league titles, the FA Cup and two League Cups.
"When I first came to Manchester I felt like when I was in Barcelona at Espanyol; we were not the main club in the city," said Zabaleta.
"We can't say the same thing now. I remember at that time Sir Alex Ferguson and the 'noisy neighbours'. For me, we couldn't let people say something like that about us.
"I thought if they call us the noisy neighbours then let's make more noise in the city, let's prove that we are a club that is looking to achieve important things, maybe one day be the main club in the city and take this club forward to the next step to make this club better."
Ivory Coast midfielder Yaya Toure looks likely to be offered a new contract by Manchester City.