A third candidate, Simba Makoni, a former finance minister, is also in the race.
Amid the rumours and suspicions generated by the long wait for results, Tsvangirai's Movement for Democratic Change (MDC) on Tuesday denied reports that it was in talks with the ruling Zanu-PF about a handover of power.
"I can confirm that there are no talks with Mugabe," Nelson Chamisa, MDC spokesman, said.
The Associated Press news agency reported that sources close to the two rivals had told them that advisers to Mugabe and Tsvangirai were meeting after the president was told he was trailing far behind the opposition leader.
Mugabe was told there could be an uprising if he were declared the winner, it reported.
Tendai Biti, secretary-general of Tsvangirai's party, told Associated Press that the report was "rubbish".
Slovenia, which holds the European Union's rotating presidency, urged Mugabe to step down.
"If Mr Mugabe continues, it will be a coup d'etat," Dimitrij Rupel, Slovenia's foreign minister, said after addressing the European Parliament.
"I hope he is on his way out, most Europeans think this way."
The MDC has repeatedly stressed that it is confident that it has won both the presidential and parliamentary contests.
Marwick Khumalo, head of the Pan-African Parliament observer mission, told South African radio on Tuesday that leading members of Mugabe's party were contemplating defeat with trepidation.
"I was talking to some of the big wigs in the ruling party and they also are concerned about the possibility of a change of guard," Khumalo said.
"Zanu-PF has actually been institutionalised in the lives of Zimbabweans, so it is not easy for anyone within the sphere of the ruling party to accept that 'Maybe we might be defeated or might have been defeated."'
Meanwhile, the United States has urged the government to release the election results quickly and respect the people's will.
"We want to see the presidential vote count be released as soon as possible," Tom Casey, US state department spokesman, said.
"Delays in that vote counting and delays in the release of the results are troubling, certainly given all the problems that we noted prior to the election."
Results have been coming in at a snail's pace from the parliamentary elections, and the MDC has accused the electoral commission of sitting on results in an attempt to fix the outcome.
With results from 131 of Zimbabwe's 210 constituencies announced, the MDC had won 67 parliament seats while Zanu-PF trailed slightly with 64 seats.