Baker has been contracted to the West Midlands side since penning professional terms in 2008, and 122 appearances have been made during that period.
However, his most consistent season arguably came on loan at Bristol during the 2015-16 campaign and it appears that the former England Under-21 international is close to a second spell at Ashton Gate.
According to the Birmingham Mail, City boss Lee Johnson wants the 26-year-old to feature in his backline as the Robins look for an improved year at the second tier.
Only a strong end to the season ensured that City avoided relegation to League One, and Baker will allegedly return to the club should he pass a medical.
Ahead of their final pre-season friendly this week, City have made just two summer signings.