The former Barcelona player is being linked with a return to La Liga five years after he left Camp Nou.
Brazilian newspaper O Globo claim that the player held talks with his agent Roberto Assuncao regarding a move away from Manchester City.
Nery Castillo's arrival at The City of Manchester Stadium on loan from Shakhtar Donetsk has also put Geovanni's future in doubt.
Assuncao has confirmed that Liga strugglers Depor are in pole position to sign the South American.
"Geovanni being transferred to Deportivo would be a good thing," the agent said. "He would be delighted to play at El Riazor.
"We have been in contact with the club for a week.
"He has a European passport, which would make the deal easier; it could happen as a loan or a transfer."