The opposition have said figures released by the commission since Tuesday's vote were "fictitious" and that "confidential sources" within the commission had provided figures showing Odinga had a large lead in the race.
Kenya's opposition coalition on Friday said it would accept the result of this week's presidential vote if the election commission granted it access to see raw data on its computer servers.
The move is a significant climb-down from its previous position when the coalition rejected figures released by the commission and said its candidate Raila Odinga should be declared president.
Many Kenyans feared the dispute would lead to violent protests after more than 1,000 people were killed following the contested 2007 election.
Provisional results from all but a few polling stations show President Uhuru Kenyatta with a lead of 1.4 million votes as he vies for a second and final five-year term. Odinga has lost the last two elections, claiming fraud in both cases.
"If they can open those servers, and we all look at it, we are prepared to accept the results of what is contained in those servers," James Orengo, chief election agent for the NASA opposition coalition, told reporters.
Call for access
Orengo also called for other candidates and observers to be given access to the servers so there could be a transparent audit of data from 41,000 polling stations across the country.
Yakub Guliye, election commissioner in charge of information technology, said the opposition had not made a formal request.
"It was a verbal request and we don't act on verbal requests. We are yet to receive a formal request," he told reporters. Normal procedure calls for the commission to release final results after cross checking its electronic tally with paper forms.
Odinga's camp has said figures released by the commission since Tuesday's vote were "fictitious" and that "confidential sources" within the commission had provided figures showing Odinga had a large lead in the race.
The election commission rejected these claims, pointing out they contained basic mathematical errors.
Murithi Mutiga, senior Kenya analyst at the think-tank International Crisis Group, said the opposition had softened its stance.
"It does look like a climb-down from their initial very strident claims they had won," he said. "The decision by NASA to declare their own tallies had sowed a lot of tensions but now their call for the IEBC to open up their servers and open it up to scrutiny ... might help calm tempers."
Beefed up security
Police had beefed up security across much of Kenya - particularly in opposition strongholds in the west and parts of Nairobi - in anticipation of the election commission announcing the election result on Friday.
At an international conference center, ruling party supporters sang "Today is our day, God is good" as the president arrived to address them. Some waved bread, showing there would be no "half loaf" - slang for a coalition government with the opposition.
Many Kenyans have feared a repeat of the violence that followed the 2007 contested election, when about 1,200 people were killed and hundreds of thousands displaced when political protests sparked ethnic killings.
Kenya is the leading economy in East Africa and any instability would be likely to ripple through the region.
Earlier on Friday, 13 out of 20 stocks on the main the Nairobi Securities Exchange NSE-20 Share Index were up, partly helped by the peaceful elections.
Odinga is a member of the Luo, an ethnic group from the west of the country that has long said it is excluded from power. Kenyatta is from the Kikuyu group, which has supplied three of four presidents since Kenya gained independence from Britain in 1963.
International observers have given the thumbs-up to the vote and U.S. Ambassador Robert Godec issued a statement on behalf of the diplomatic community calling for any complaints to be channelled through the courts, not street protests.
"If there are disputes or disagreements, the Kenyan constitution is very clear on how they are to be addressed. Violence must never be an option," he said on Friday.
"Now is the time for leaders, across the political spectrum, to demonstrate their commitment to (the) constitution and to the institutions it creates and the values it sets out."
As well as a new president, Kenyans also elected new lawmakers and local representatives. Some of those races have also been disputed, leading to violence in Garissa and Tana River counties.
Four people were killed in election-related violence on Wednesday but demonstrations have otherwise mostly been brief and isolated.
In Kibera, a Nairobi slum, some Odinga supporters conceded that the margin of Kenyatta's lead looked unassailable.
"The vote is clear. It was a very good election, there were no problems and now he must concede," Mohammad Amber, a 40-year-old engineer who voted for Odinga, told Reuters. "He must move on. Even in football, there are winners and losers."