Argentine ace Arca is set to go under the knife on his injured knee, while captain Thirlwell suffered a fractured skull in the reserves on Tuesday.
It means that Sunderland boss Mick McCarthy will be without the key duo for the rest of the season and the play-offs.
"Julio will have an operation on Tuesday and that's him done," McCarthy told the club's official website.
"It's something that has been niggling at him for some time. He wanted to play on but he couldn't. It's a blow for us and a blow for him.
"Paul's out of hospital and I've spoken to him, he'll be okay. It's a bad injury and he won't play again this season.
"It's sad for him and Julio because they won't be involved again this season and that includes the play-offs."
