Their presence in Qatar had severely strained Doha's relations with Egypt as well as Saudi Arabia and the UAE, all of which view the more than 85-year-old Islamist movement as a threat.
The expulsion threatens to further isolate the group, which rose to power in Egypt through a string of post-Arab Spring elections but suffered a dramatic fall from grace during Morsi's divisive year in office.
"I decided to move outside of the beloved Qatar ... So as not to cause any annoyance, embarrassment or problems for our brothers in Qatar," Ghoneim said in a video message posted on his official Facebook page.
Darrag, in a statement posted on his page, said, "we value the role of Qatar in supporting the Egyptian people in its revolution against the coup. We understand well the conditions it is facing in the region."
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According to Rassd, a news agency affiliated with the group, Hussein is among those who will be leaving the country. The agency said they will be searching for another base in exile, possibly Turkey.
Qatar was a close ally of Morsi, and relations between Cairo and Doha have been tense since his ouster, with Egypt accusing Qatar of backing the Brotherhood.
Egypt has accused Qatar's Al-Jazeera network and its offshoots of serving as a mouthpiece for the group in its campaign against the government.