Sane has spent time in America undergoing his rehabilitation in the company of City staff after breaking down early in the first half of the Community Shield against Liverpool at Wembley in August.
Although initially ruled out until March, there is growing optimism he could return to action in February and play a part in the final months of the campaign.
Sane has faced questions in the past about his attitude and struggled for starts during the second half of last season after falling out of Guardiola’s favour but his potential departure would be a severe blow to City. He has 39 goals in 134 appearances for City since his projected £42m move from Schalke in 2016 and his absence has been felt at times this season.
Oyarzabal is believed to one potential replacement for Sane. The 22-year-old Spain international – who is contracted to Sociedad until 2024 – reportedly has a €75m (£64m) release clause in his contract with the Basque club.
City’s director of football, Txiki Begiristain, is a Basque who has strong ties with Sociedad, with whom he started his professional career, emerging from the club’s junior to spend six years in the first team before a move to Barcelona in 1988.