The call comes a short time after Obasanjo's eldest son said he was misquoted by a newpaperman when he derided the activities of Vice President Atiku Abubakar.
For his part, the Vice President said Obasanjo had played fast and loose with the possibility of seeking a third term, which Obasanjo says would depend on whether or not the 1991 Nigerian constitution is amended to allow it. Currently, he is limited to two terms, and as in American politics, the Vice President is always considered a front-runner for the presidency.
While many Western leaders have publicly and privately indicated displeasure with Obasanjo's weighing of a third-term bid, ERHC On The Move has opined that Nigeria needs a strong leader with a strong anti-corruption message, and none of the potential candidates for the presidency seem to fit that bill well. Most have aggressivekly fought implementation of the anti-corruption campaign.
Here is the story:
Obasanjo should resign, VP says
Vice-President Atiku Abubakar has helped Obasanjo win two elections.
The row between Nigeria's leaders has intensified, with the vice-president urging President Olusegun Obasanjo to resign "for breaking the constitution".
Vice-President Atiku Abubakar opposes moves to let Mr Obasanjo seek a third term in office.
His comments come after he was told to step down by Mr Obasanjo's spokesman.
Mr Obasanjo has not publicly said whether he wants to remain in office but both men are believed to want to contest elections due next year.
They have been president and vice-president since the end of military rule in 1999.
Mr Obasanjo's supporters have been campaigning for the constitution to be changed, in the face of vigorous opposition from various groups.
Hostility between Mr Obasanjo and his deputy has been brewing beneath the surface for some time, fuelled by political ambition, but has now come into the open, the BBC's Alex Last reports from Abuja.
Mr Obasanjo's second term in office ends next year.
"The call for resignation should be directed at the president... for pursuing an agenda that is subversive of the constitution, and the will of the majority of Nigerians," said the statement signed by Mr Abubakar's spokesman.
"For the avoidance of doubt, the vice president believes tenure elongation is morally wrong and a breach of the constitution, which he took the oath to defend," it said.
Presidential spokesman Femi Fani-Kayode replied that Mr Obasanjo had not said he wanted to seek a third term and so had not broken the constitution.
Mr Fani-Kayode had earlier said that if Mr Abubakar was unhappy in government, he should do the right thing and step down.
Mr Abubakar on Wednesday night publicly declared his position for the first time, attending a meeting of senior figures opposed to changing the constitution.
He said he was prepared to take this stand because anything the government did to him could not be worse than his last three years in office.
Former military leader Muhammadu Buhari, who lost the 2003 elections to Mr Obasanjo, attended the meeting, along with MPs and both serving and former state governors.
State security agents said the group could not hold the talks at the Abuja Sheraton hotel because they had not sought police permission.
Instead, they moved to a government office.
The issue has divided the ruling People's Democratic Party and Nigerian public opinion.
Relations between the president and his vice-president have been strained since last August, when President Obasanjo publicly accused his deputy of disloyalty.
On Wednesday Mr Obasanjo's spokesman told the BBC that the president would consider whether or not to stand if the constitution was changed.
The National Assembly is due to consider more than 100 proposed constitutional amendments, including whether to extend the limit on a president's term in office from two to three terms.
Opponents of the constitutional change argue that the presidency needs to rotate among people from different regions and ethnic groups.
Mr Obasanjo is a Christian from the south-west while Mr Abubakar is a Muslim from the north.
Recently, a majority of Nigeria's state governors agreed that a constitutional review was necessary within the life of the current administration.