Vale Boss Dean Glover told the Stoke based The Sentinel newspaper: "He's a name that has been brought up before and he's somebody we might be interested in if he's available.
"I've not made contact with Coventry, or with Chris himself, and I think he's away with Trinidad at the moment.
"I don't know if we will make contact and I don't know what his situation is at Coventry, although he has not been in favour there."
Birchall, who helped City win the Birmingham Senior Cup in his first season, is available for loan or a permanent move and is currently on international duty with Trinidad & Tobago, who play Cuba on Wednesday.