The president needs parliament's approval to sack the prime minister, according to Somali transitional government's federal charter.
Hussein, who became Somalia's prime minister in November 2007, has in the past few months been in dispute with Yusuf over efforts towards a peace deal with the Alliance for the Re-liberation of Somalia (ARS), the country's main opposition group.
“I have heard about the president’s appointment of a new prime minister. The president has repeatedly violated Somalia’s charter and the interim government’s charter, and this is not new," Hussein told Al Jazeera.
“I reject the appointment of a new prime minister as the president does not have the power to dismiss the prime minister; only the parliament can dismiss or withdraw confidence in the prime minister.
"The president can not be above the law or above the constitution, and he can not do whatever he wants."
Mustafa Duhullow, Somalia's agriculture minister, dismissed Yusuf's decision as a "personal wish that will not have legal effect."
"Hussein is the legitimate prime minister of Somalia approved by the parliament, but the president has the right to nominate whoever he wants," Duhullow told the news agency AFP.
"This is a desperate measure by the president to interrupt the ongoing peace talks between the TFG [Transitional Federal Government] and the opposition," he said.
Aden Mohamed Nur, the parliamentary speaker, said the vote results on Monday showed that 143 members of parliament had "recognised the existence of the government, 20 rejected it and seven abstained".
Kenya imposed sanctions on the Somali president and his family on Tuesday.
Moses Wetangula, Kenya's foreign affairs minister, said Yusuf was an obstacle to peace. The sanctions include a travel ban and freezing any assets in Kenya.
Somalia has been without an effective government since the overthrow of Mohamed Siad Barre, the former president, in January 1991.