The full-back made 37 appearances for the Black Cats last year but will have to watch from the sidelines during the campaign's opening matches.
Bardsley first picked up the ankle knock at the start of pre-season training and has been forced to sit out Sunderland's summer fixtures.
O'Neill said: "Stephane has done a little bit of training this morning, and hasn't felt any effects. Wes Brown is feeling a bit better but obviously getting a bit annoyed that injuries have plagued him.
"David Vaughan's had an operation on his side to cure a problem and he's making good progress out there on the training field, but hasn't joined in with us yet."
Sunderland begin their season with an away trip to Arsenal on Saturday.