Morsi served as chairman of the Muslim Brotherhood's Freedom and Justice Party, running as its candidate for president. He resigned from the Muslim Brotherhood and FJP after he was declared the winner of a runoff election against former Prime Minister Ahmed Shafiq last weekend.
Morsi is expected to take the oath of office this week.
Essam El-Erian, vice chairman of FJP, denied claims that he was taking over from Morsi.
"Erian assured that the selection of a party leader, according to the FJP's regulations and internal charter, is the prerogative of the FJP's General Conference, and that so far no date has been set for the convening of this conference," a statement from the Muslim Brotherhood read.
There is no formal constitution that defines the role of the Egyptian president. Members of a constitutional panel met Wednesday to discuss whether post-revolution Egypt would be a presidential, parliamentary or mixed political system, Egyptian news service al-Ahram reports.