Nahid Islam in a video post in social media said the student protest leaders had already talked with Yunus, who consented to take over considering the situation of the country.
Bangladesh's figurehead president and its top military commander said an interim government would be formed soon.
Yunus, who called Hasina's resignation the country's "second liberation day", faced a number of corruption accusations and was put on trial during the former prime minister's rule.
He received the Nobel Peace Prize in 2006 after he pioneered microlending, and said the corruption charges against him were motivated by vengeance.
Violence just before and after Sheikh Hasina's resignation left more than 100 people dead. (AP PHOTO)
Islam said the student protesters would announce more names for the government, and it would be difficult for the current leadership to ignore their choices.
Hasina resigned and fled the country after weeks of protests against a quota system for government jobs descended into violence and grew into a broader challenge to her 15-year rule.
Thousands of demonstrators stormed her official residence and other buildings associated with her party and family.
Her departure threatened to create even more instability in the densely populated South Asian nation that is already dealing with a series of crises, from high unemployment and corruption to climate change.
The streets of Dhaka appeared calmer on Tuesday, with no reports of new violence, and flights resumed at the main airport in Dhaka.
Violence just before and after Hasina's resignation on Monday left at least 109 people dead and hundreds of others injured, according to media reports.
In the southwestern district of Satkhira, 596 prisoners and detainees escaped from a jail on Monday night after an attack on the facility, the United News of Bangladesh agency reported.
Thousands of people poured into the streets to celebrate Hasina's resignation. (AP PHOTO)
The military chief, General Waker-uz-Zamam said he was temporarily taking control of the country, and soldiers tried to stem the growing unrest.
Mohammed Shahabuddin, the country's figurehead president, announced late on Monday after meeting with Waker-uz-Zamam and opposition politicians that parliament would be dissolved and a national government would be formed as soon as possible, leading to fresh elections.
Speaking after the embattled leader was seen in television footage boarding a military helicopter with her sister, Waker-uz-Zaman sought to reassure a jittery nation that order would be restored.
The main opposition Bangladesh Nationalist Party on Tuesday urged people to exercise restraint in what it said was a "transitional moment on our democratic path".
The United Nation's human rights chief, Volker Turk, said the transition of power in Bangladesh must be "in line with the country's international obligations" and "inclusive and open to the meaningful participation of all Bangladeshis".
Hundreds of thousands of people poured into the streets waving flags and cheering to celebrate Hasina's resignation.
Protests began peacefully in July as students demanded an end to a quota system for government jobs they said favoured those with connections to the prime minister's Awami League party.
Prime Minister Sheikh Hasina landed at a military airfield near New Delhi after leaving Dhaka. (AP PHOTO)
But amid a deadly crackdown, the demonstrations morphed into an unprecedented challenge to Hasina.
Waker-uz-Zaman promised the military would investigate a crackdown that had left nearly 300 people dead since mid-July, some of the country's worst bloodshed since the 1971 war of independence and which had fuelled outrage against the government.
Nearly 100 people, including 14 police officers, were killed on Sunday, according to media reports.
At least 11,000 people have been arrested in recent weeks.
Hasina landed at a military airfield near New Delhi on Monday after leaving Dhaka and met India's national security adviser Ajit Doval, the Indian Express newspaper reported.
Hasina was taken to a safe house and would likely travel to the United Kingdom, the report said.
The 76-year-old was elected for a fourth term in a January vote that was boycotted by her main opponents.
Thousands of opposition members were jailed before the poll, and the US and the UK denounced the result as not credible, though the government defended it.
