Bardsley sustained the injury early in pre-season and is set to miss 'a number of weeks'.
Sessegnon returned to training on Tuesday at the Academy of Light after an ankle complaint and O'Neill hopes he will be back in action soon after the Black Cats' opening game away to Arsenal this weekend.
"Stephane has done a little bit of training this morning and hasn't felt any affects," O'Neill said.
"He hasn't been able to do that much but in the course of the next few weeks in terms of the sort of fitness level that he can attain, we hope he gets there pretty quickly.
"Wes Brown is feeling a bit better but obviously getting a bit annoyed that injuries have plagued him.
"David Vaughan's making progress and he has had an operation on his side to cure a problem. He's making good progress out there on the training field, but hasn't joined in with us yet."
English Premier League
Liverpool forward Fabio Borini will not be moving to Serie A outfit Napoli, according to his agent.