Hearts interim head coach Jon Daly says Celtic boss Brendan Rodgers' comments about his club were "unacceptable".
And the Northern Irishman questioned how the club recruited.
"I'm extremely disappointed," said Daly. "The fact that he's commenting on the structure of our football club, when he knows nothing about it."
On Friday, Rodgers praised Cathro, who worked under director of football Craig Levein, as a "very good coach" who "wanted to bring in his own ideas".
The Celtic boss added that while watching Hearts "it always looked a little bit confused in terms of what he wanted and maybe what other people wanted.
"You know, you're trying to play football and you're bringing in players who play a direct game. So, it makes me ask about where the players are coming from? And if they're his?"
Following Saturday's defeat at Celtic Park, Daly told BBC Scotland: "I can understand the comments made about a manager losing his job because I'd be the very same. It's never nice to see someone lose their job but to comment on the recruitment of the players, to say that they don't fit the system, it's poor for me.
"You look at the players we've brought in - Michael Smith, he's a wing-back; Ash Smith-Brown, wing-back; [Rafal] Grzelak, wing-back; [Kyle] Lafferty, striker; [Isma] Goncalves, last year to play on the left-hand side of a front three; [Christophe] Berra to play in a back three. For him to come out and make a comment like that, it's just farcical, to be honest.
"I've followed Brendan's career for a long, long time. I've got a lot of admiration for him and I've never heard him make a comment like that when he was at Liverpool, commenting on Man United's structure or Chelsea's structure or Man City's structure.
"He's obviously come up to Scotland and think he's the biggest fish in the biggest pond, with the best squad, with the best budget and he thinks he can comment on other people's teams, other people's structures or clubs and it's unacceptable.
"He wouldn't have done it in the Premiership [in England] so I don't see why now he all of a sudden thinks he can start doing it."
In his post-match interview, Rodgers said: "I was purely asked a question in relation to the operation there. It's no problem. I thought Jon organised the team well today. They were good and compact.
"My feeling was for support for young Ian Cathro, who I felt was dealt harshly with. But I've got no issue, there's no problem. I hope Jon goes on and gets the job.
"I'll always be open and say what I feel. That doesn't change. I say and believe in what I say 150%."
Asked if there was perhaps a an unwritten rule that managers do not discuss other managers or clubs, Rodgers replied: "Yeah, well, he hasn't become a manager yet so wait until he becomes a manager and then he'll know the rules of the game then."