Despite producing a strong performance in the 1-1 draw with Sweden on Monday, Ireland are likely to need a win in one of their remaining games if they are to progress to the knockout stages, starting with Belgium on Saturday.
O'Neill's men then take on Italy in their last group game of the tournament, with the Azzurri arriving in France under a fanfare of low expectation.
Nevertheless an impressive performance from the four-time world champions during the 2-0 win over heavily-fancied Belgium means they have gone some way to proving any doubters wrong.
O'Neill insists he is not remotely surprised to see Italy being talked up once again, adding that they still have the ability to make a mark on the international stage.
According to RTE, he told reporters: "This idea that Italy are a has-been side has been totally negated. Italy are a tournament team, we know this from years and years and years of experience.
"They qualify for competitions because they are a strong, strong footballing nation. They have got very, very good players now and they will be hard [to beat]."
For now, Ireland's attention will turn to Belgium on Saturday, and the former Celtic boss says they too will be a danger to his side's tournament ambitions.
He added: "Belgium have got a real set of individual players - it's whether they go and gel as a side or not. But they have individuals capable of taking you on and beating you and they play in top-class company every single week."
Ireland could be without Jonathan Walters for the trip to Bordeaux this weekend, amid fears the Stoke City ace has further aggravated an Achilles tendon injury after limping out of Monday's match midway through the second half.
Walters later admitted he was in pain from the start of proceedings in Paris, and O'Neill says he is unlikely to start this weekend.
"Jon Walters would be the main concern," he added. "I'd say he would be pretty doubtful for the second game coming up."