"They are saying that they are confident that Karzai will win the race in the first round with more than 50 per cent of the vote," he said.
But Ahelbarra added it "remains to be seen whether Karzai will accept to work with a prime minister and share power. He has been saying up to this moment that he is the man of consensus".
In another opinion poll in May, Karzai received 31 per cent support, while Abdullah only received seven per cent.
The surveys, which were the first since the presidential campaign began in mid-June, were published as Karzai urged Afghans to take part in the vote.
"The elections will pass peacefully ... the enemies of Afghanistan will try to create some chaos, [but] you don't bother about it," he said at a high school on Tuesday.
The Taliban have vowed to disrupt the August 20 presidential elections, and have stepped up their attacks in recent weeks.
On Monday, six suicide bombers attacked the governor's compound and elections offices in Logar province, south of the capital, Kabul.
A United Nations report has said that security fears in many parts of Afghanistan were hindering preparations for the presidential election to be held on August 20.
The report, compiled by the UN mission in Afghanistan and Afghanistan's independent human rights commission (AIHRC), said insecurity had "severely limited freedom of movement and constrained freedom of expression for candidates".
In some areas there have been no campaign rallies and concerns remain that voting may not be possible in some parts of the country.
A final list of areas where voting will be able to take place is expected to be published on August 15.
The elections are crucial for Western efforts to stabilise Afghanistan, where more than 90,000 foreign troops are fighting the Taliban.