Morsi last week accused private media of fanning violence and argued that it was being used for political aims.
But Yassin denied that the prosecutor's office was operating at the behest of the presidency to go after Morsi critics, saying it has also interrogated and sentenced Islamists.
"There is no contact between us and the presidency. ... Just like we moved against someone who insults Christianity, we moved against someone who is accused of insulting Islam," he said.
Youssef is the host of the weekly political satire show known for his skits lampooning Morsi and Egypt's newly empowered Islamist political class. But he also mocks the opposition and the media.
The fast-paced show has attracted a wide viewership, while at the same time earning its fair share of detractors. Youssef has been a frequent target of lawsuits, most of them brought by Islamist lawyers who accused him of "corrupting morals" or violating "religious principles."
Youssef frequently imitates Morsi's speeches and gestures. He has fact-checked the president, and in one particularly popular episode earlier this year, he played video clips showing remarks by Morsi, made in 2010 before he became president, calling Zionists "pigs."
The remarks caused a brief diplomatic tiff with the U.S. administration, and Morsi had to issue a statement to defuse the flap.
In his last episode this week, Youssef thanked Morsi for providing him with so much material.
Youssef has also made regular jokes about comments by Islamic clerics and Islamic stations TV presenters, exposing contradictions between their comments and public speeches and what he considers the spirit of Islam.