In a televised speech, Burhan said the country's two-year transitional period will end with free and fair elections and vowed to put on trial those involved in the killing of the protesters as well as corruption under the old regime.
Military bulldozers and tractors moved to remove roadblocks erected by protesters near the headquarters but were forced back by protesters, according to eyewitnesses.
They say the bid to move them is proof that the security forces can not be trusted and shows why Sudan so badly needs an accountable civilian government.
The more than 5,000 protesters led by the Sudan Professionals Association linked hands and formed a ring around the sit-in zone they have been occupying since last December.
Thousands remained encamped outside Khartoum's army headquarters to keep up pressure on a military council that took power after ousting Bashir on Thursday.More news: 'Significant fire' underway at Notre Dame Cathedral in Paris
"It is time for the transitional military council and all other parties to enter into an inclusive dialogue to effect a transition to civilian rule", they said.
The deputy leader also told the USA envoy how the military council would "preserve the security and stability of the country".
"We hope that everyone will head immediately to the areas of the sit-in to protect your revolution and your accomplishments", the SPA said.
The protest outside the compound, which also includes the intelligence headquarters and the presidential residence, began on April 6, after more than three months of protests triggered by a deepening economic crisis.
Britain's ambassador to Sudan, Irfan Siddiq, said he had met the deputy head of the transitional military council, Mohamed Hamdan Dagalo, and had "asked for clarity on whereabouts of former President Bashir and other senior former regime figures".
"We want to set up a civilian state based on freedom, justice and democracy", a council member, Lieutenant General Yasser al-Ata, told members of several political parties.More news: Man accused of jumping from plane onto Phoenix Sky Harbor tarmac
Below are their key demands, which the umbrella group Alliance for Freedom and Change says must to be met for the sit-in to end.
Sudan's military council is headed by General Abdel Fattah al-Burhan, who took power after his predecessor General Awad Ibn Auf stepped down on Friday.
Protest leaders have called for the feared intelligence agency, whose chief Salah Gosh resigned on Saturday, to be restructured and for NISS officials to face prosecution.
Talks between protest leaders and Sudan's new rulers were followed on Sunday by a meeting between Washington's top envoy to Khartoum, Steven Koutsis, and the military council's deputy.
The SPA has also called for the confiscation of properties belonging to the ousted president's National Congress Party and the release of soldiers who sided with their movement.More news: The Shocking loss Meghan Markle has suffered after marrying Prince Harry