The MDC has been warned against declaring itself the winner
Zimbabwe's government and electoral chiefs have warned the main opposition MDC it should not declare an early victory in the presidential poll.
The MDC, which has repeatedly expressed fears of rigging, has started to quote unofficial returns, saying it has 67% of the vote so far and "has won".
The information minister accused the MDC of "speculation and lies".
An African poll observer said most votes had been counted, and delays in releasing results were causing anxiety.
Incumbent President Robert Mugabe was facing a challenge from the MDC's Morgan Tsvangirai and the independent Simba Makoni.
House of Assembly, Senate and local elections were held on the same day. Officials say the final results may not be known for a few days.
The secretary general of the Movement for Democratic Change (MDC), Tendai Biti, says the party has returns from one-third of polling stations.
Mr Biti says they show a 67% vote for the MDC in the presidential vote, with the party also winning most parliamentary seats in Harare and Bulawayo.
Mr Biti said the MDC was also ahead in some rural areas, including Mr Mugabe's home province of Mashonaland West.
"We have won this election," said Mr Biti. "This trend is irreversible."
BBC Southern Africa correspondent Peter Biles says the MDC's claims are based on partial, unofficial results.
The more slowly-counted votes from the rural areas, where President Mugabe has always had majority support, may decide the final outcome.
Mr Biti also questioned why it was taking so long for the results to be announced, as returns have been posted outside polling stations.
BBC contributors say opposition activists have been celebrating in the towns of Bulawayo, Mutare and Masvingo.
A spokesman for Mr Makoni told the BBC News website that the MDC had "swept the board" in the parliamentary election, with several ministers losing their seats.
But Zimbabwe's chief elections officer expressed concern "that some stakeholders have gone on to announce purported results... when in fact the results are being verified and collated".
The MDC says the commission was appointed by Mr Mugabe and is not to be trusted.
Information Minister Sikhanyiso Ndlovu said: "Biti and the MDC are famous for speculation and lies peddling in the country and causing unnecessary havoc here."
The state-run Sunday Mail quoted the ministry's secretary, George Charamba, as saying that if Mr Tsvangirai declared himself president "it is called a coup d'etat and we all know how coups are handled".
Mr Biti said the MDC was just "protecting its vote" and would not make the "mistake" of the 2002 and 2005 elections when it did not claim victory.
A British Foreign Office minister, Mark Malloch-Brown, said it was "quite likely" that President Robert Mugabe had lost the election in Zimbabwe, despite "massive pre-election day cheating".
A candidate needs more than 50% in the presidential vote to avoid a run-off in three weeks' time.
Across the country on Saturday, there were reports of voters not being allowed to cast ballots - either because their names were not on the voters' roll or because they were trying to vote in the wrong ward.
The opposition feared many voters would be intimidated and stay at home.
HAVE YOUR SAY
But many voters told the BBC the system had worked efficiently and the atmosphere was good.
The elections will no doubt be free and fair. The allegations being peddled are unfounded
Sosten Musiniwa, Harare
After voting in Harare, Mr Mugabe, in power since 1980 and seeking a sixth term, dismissed opposition concerns, saying: "We don't rig elections. I cannot sleep with my conscience if I have rigged."
The MDC says it is fighting to save Zimbabwe's economy.
The country has the world's highest inflation rate, at more than 100,000%, and just one adult in five is believed to have a regular job.
US Secretary of State Condoleezza Rice said on Sunday that "the Mugabe regime" was "a disgrace to the people of Zimbabwe and... to the continent of Africa as whole".
The chiefs of Zimbabwe's police, army, prison service and intelligence services warned on Friday that violence after the polls would not be tolerated.