Following a meeting in Khartoum with Omar al-Bashir, the Sudanese president, Konare said: "I can confirm today that we have received sufficient commitments from African countries that we will not have to resort to non-African forces."
Last month's UN Security Council resolution allows for a force of 20,000 peacekeepers and 6,000 civilian police.
Konare later said that outside contingents would be needed only if African countries did not follow through with their commitments.
He said: "Non-African forces would be needed only in case the African countries would not be in a position to provide the number of troops agreed-upon."
The AU chairman said representatives from the AU and the UN would meet in New York in September to discuss the hybrid force.
Disagreements over the composition of the mission were a major reason that the authorisation was delayed for months despite mounting pressure for Sudan to stop the violence that has killed more than 200,000 people and displaced 2.5 million in the past four years.
Konare gave no details about which countries had pledged forces, but several African nations have made announcements regarding troops.
In addition to the AU forces currently in Darfur that are expected to stay on, Nigeria, Malawi and Rwanda have offered to deploy another battalion each, about 2,400 troops in total, and Senegal has said it will triple its contingent to 1,600.
Burkina Faso, Cameroon, Egypt and Ethiopia also have pledged to contribute troops or add to current contingents for a joint force.