He vowed to recover billions of dollars "stolen" under previous administrations.
Mr Buhari held talks with state governors over the lack of money to pay government workers their salaries.
He and his All Progressives Congress won elections in March, ending some 16 years of rule by the Peoples Democratic Party.
He was sworn in less than four weeks ago, taking over from Goodluck Jonathan, who had been in office since 2010.
"The days of impunity and lack of accountability are over," he told state governors.
"The next three months may be hard, but billions of dollars can be recovered, and we will do our best," he added.
In an earlier briefing with journalists, he said it was a "disgrace" that government workers had not been paid for months.
African news updates
"This bad management we find ourselves in - we really need your help to protect us from people before they march on us," he said.
Vice-President Yemi Osinbajo estimates that Nigeria's debts stand at about $60bn (Â£38bn).
However, former Finance Minister Ngozi Okonjo-Iweala has rejected the claim, saying the debt was much lower and most of it was incurred by states, rather than the federal government, AFP news agency reports.
The president won the elections on a promise to tackle corruption, and to lead Nigeria to greater prosperity.
Nigeria is Africa's biggest oil producer, but most of its citizens are poor.
Nigeria's treasury is "virtually empty", President Muhammadu Buhari has said.