The Muslim Brotherhood said it was clear its candidate Mohammed Mursi had won and criticized army-led plans to curb presidential powers.
But a spokesman for Ahmed Shafik, deposed dictator Hosni Mubarak's last prime minister, said his candidate had won a majority.
Shafik's campaign spokesman, Ahmed Sarhan, told a televised news conference that he won 51.5 percent of the vote and that Mursi's claim of victory was "false."
"General Ahmed Shafik is the next president of Egypt," said Sarhan. He said Shafik won some 500,000 votes more than Mursi.
The Brotherhood said Mursi had a clear mandate, having received 52 percent of the vote compared to Shafik's 48 percent.
"We are speaking of facts and documents and not indicators or speculation. That is what is different from what the other party says to you," said Yasser Ali, spokesman for Mursi campaign, said, issuing numbers that the group compiled from their records of the count.
The country's highest electoral body is due to confirm the result on Thursday and Ali said he was confidence it would back his polling.
"There is no shred of doubt that these numbers will be the numbers that the election committee will announce," he said. "There may be slight changes after appeals from both parties but we are confident in what we say."
The rival claims carry the potential for a new chapter of unrest at a time when opposition already is growing against a "constitutional declaration" announced by the military on Sunday which robbed the next president of many powers and gave the generals who succeeded Mubarak last year legislative powers as well as control over the process of drafting a constitution.
The Brotherhood has called for a mass rally on Tuesday evening against the military council, with thousands expected to take to the streets.
But Ali stressed that the party were not seeking further confrontation with the powerful ruling military council.
"We do not seek any confrontation with anyone and no one in Egypt wants confrontation," he said.
"There has be dialogue between national forces and the people alone must decide their fate."
(Al-Akhba, AFP, AP)
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