In the city of Cochabamba, the scene of violent clashes, Reuters journalists reported seeing police officers protesting on the roof of their headquarters in an apparent act of disobedience against the government.
Defense Minister Javier Zabaleta said a "police mutiny occurred in a few regions", but he rejected the idea of a military intervention "at this time".
Hundreds of opposition supporters crowded around police stations in La Paz and in the northeastern city of Trinidad to urge police there to reject Morales.
On Wednesday he flew to La Paz, Bolivia's capital, with a resignation letter for Mr Morales to sign and a bible "in order that God returns to the palace".
Mesa, who came in second in the October 20 election, joined former presidents Jaime Paz Zamora and Jorge Quiroga in urging the military not to put down the protests.
At a news conference later in the day, Morales called an urgent meeting with the four political parties represented in parliament.
"The government is now obliged to manage the crisis without the police and attempt to offer a solution that addresses the large share of Bolivians who have no confidence in the election outcome", he told the Observer.
Morales, Latin America's longest-standing leader, won the election on October 20 but the vote count had been inexplicably halted for almost a day, sparking allegations of fraud and leading to protests, strikes and road blocks.
There was no immediate comment from either Camacho or Mesa.
Opponents challenge an official count that showed Morales winning with 47% of the vote and a margin of just over 10 percentage points over his nearest competitor - enough to avoid the need for a runoff against a united opposition. But a 24-hour lapse in releasing vote results raised suspicions among opposition supporters.
Morales said he would also invite worldwide organizations including the Vatican, the United Nations and the Organization of American States (OAS), which is conducting an audit of the October vote. The opposition says it will not accept the results because they were not consulted on how the process would unfold.
Morales' bid for re-election was controversial before it began. But he refused to accept the results of a referendum upholding term limits.
The country's constitutional court later ruled that term limits violated his right to run for office, and the electoral court accepted his candidacy for a fourth term.