The Black Cats suffered a 1-0 defeat to Spurs at White Hart Lane on Sunday, having already avoided relegation, but the Italian, who left the pair out of his squad for the trip to north London, was "disgusted" with his charges' lack of professionalism.
"They can't stay with me, people who behave like that," Di Canio told BBC Sport. "It was disgusting. These people can't train with me."
|PREMIER LEAGUE LATEST
|11/5||Crystal Palace are 11/5 with Bet365 to be promoted to the Premier League
"I know also someone can make a mistake - I did in the past, maybe not in that way, but every time we win or we are happy we are going to behave like this? It is ridiculous and pathetic."
Di Canio is looking to put his own stamp on the squad in the summer after replacing Martin O'Neill on Wearside in March.
"I am going to sit at the table with the chairman," he added. "Obviously if we want to have a different future in terms of results we need to bring in good players but with a professional ethic.
"Be a real professional, not a good footballer that performs 40 or 50 per cent of the potential they have got because they got blurry or because they go out.
"We have to make sure we are going to bring in not only good players but with a good professionalism that can give satisfaction to everyone."