Former Manchester United full-back Bardsley is expected to play a key role for the Black Cats again next season after making 31 Premier League appearances last term.
Meanwhile, York-born Kilgallon made just seven league appearances in the previous campaign, but has been robbed of a chance to impress in pre-season by his injury.
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O'Neill told the Shields Gazette: "Phil went over on his ankle and it's really sore. Nothing's broken, but he's done a bit of damage.
"He's very doubtful and I don't think the trip would be good for two things. One, he wouldn't be able to play and two, being back and forth on a plane would be no benefit.
"Matt had been trying to do some work himself over the summer to keep fit and he injured himself doing that. He came back with a bit of a problem, so the best thing to do is to leave him out."
O’Neill also revealed he is set to step up his search for new players as he prepares for his first full season in charge of the Wearsiders.
He added: "I haven't done anything definite. I've made the enquiries and said to people, keep us informed. But now I will start to develop things a bit more."