City have started the season in blistering fashion winning 18 out of 20 games in all competitions, but boss Pep Guardiola will not rest on his laurels and is looking to strengthen further.
Central defence could be a problem area for City, who have just lost John Stones to a hamstring injury, while Vincent Kompany continues to be dogged by injuries and has only just returned from a calf problem.
Guardiola wants cover and is reported to be targeting a move for Martinez, a player who has been on his radar since his days in charge of Barcelona.
The 26-year-old is under contract until June 2021, but Sociedad face a battle to keep hold as City prepare to test their resolve by making a bid in the new year.
There will be strong competition for his services though as Real Madrid, AS Roma and Liverpool are also said to be in the hunt.