The denial came as President Beshir and the top opposition leader and former Prime Minister, Sadeq al-Mahdi, reportedly agreed in principle to meet soon for the first time since the 1989 coup that overthrew Mr Mahdi's government.
Reports of a power struggle in Sudan followed an open confrontation between Mr Turabi and President Beshir this week.
Mr Turabi - long seen as Mr Beshir's mentor and the ideologue behind the radical Islamic government - blocked a move by the president to delay reforms that would reduce his powers.
Foreign tour postponed
The confrontation, which came to a head on Monday, was serious enough to cause Mr Beshir to postpone a foreign tour due to start that day.
However, in a BBC interview, Mr Turabi tried to downplay the rift. Asked why he had ignored the president's request for a postponement of the debate, Mr Turabi, a constitutional lawyer, took refuge in technicalities.
"I didn't ignore him. But I'm subject to regulations here. Once I distribute the order of the week to members of parliament, it just has to come through on each day of the week actually. His letter arrived too late," he said.
Mr Turabi denied that he was involved in a power struggle with Mr Beshir; he was simply using his position to develop democracy in parliament.
Important questions raised
The BBC's Africa reporter, Virginia Gidley-Kitchen says the row raises two questions: how deep Mr Turabi's rift with Mr Beshir goes and why he should wish to introduce more democracy.
"At present it seems safest to assume that the two men differ about means rather than ends - that they both still want to preserve Sudan's Islamic-inspired military government, which is currently crippled by the cost of the war with the southern rebels and the country's international isolation," she explains.
"Mr Turabi appears to think that some modest political liberalisation will enable him to kill two birds with one stone: to improve relations with the West and so gain access to Western aid, and to tempt home elements of the northern political opposition which had joined forces with the southern rebels."
Peace initiative
Mr Mahdi's Umma party said in statement that the planned meeting between him and President Beshir would help end the civil war in Sudan.
The statement said that it was eager to promote inter-Sudanese dialogue in order to end the divisions in the country.
"General Beshir is considered an important factor in the internal [political] equation," the statement said.
It added that the meeting "could provide an impetus to efforts for a comprehensive peace settlement"