Yunus, 84, Bangladesh's only Nobel laureate, was pushed for the job by the student protesters who led the campaign against Hasina. He was expected to be sworn in as chief adviser along with a team of advisers later on Thursday.
Yunus, a harsh critic of Hasina, is known as the 'banker to the poor' and was awarded the 2006 Nobel Peace Prize for founding a bank that pioneered fighting poverty with small loans to needy borrowers.
He is due to arrive in the capital Dhaka from Paris on Thursday, where he had been receiving medical treatment.
"I'm looking forward to going back home and see what's happening there and how we can organise ourselves to get out of the trouble that we're in," Yunus said before he boarded a flight on Wednesday evening.
People gather in front of the Bangladesh Nationalist Party (BNP) headquarters during a protest rally (AP PHOTO)
Hasina's dramatic exit on Monday from the country she ruled for four terms - and was reelected to a fifth in January - triggered jubilation and violence across Bangladesh, as crowds stormed and ransacked her official residence unopposed.
She fled to neighbouring India where she is taking shelter at an air base near the capital New Delhi.
The daughter of state founder Mujibur Rahman, Hasina survived the assassination of her father and most of her family in 1975. She ran Bangladesh for 20 of the last 30 years.
Student protests against controversial quotas for government jobs spiralled in July and more than 250 people were killed and thousands injured as protesters clashed with security forces and supporters of Hasina's Awami League party.
The protests were fuelled by tough economic conditions and political repression as well.
After years of strong growth as the garment industry expanded, the $US450 billion ($A690 billion) economy struggled with costly imports, inflation and unemployment and the government had sought a bailout from the International Monetary Fund.
Yunus and the Bangladesh Nationalist Party (BNP), Hasina's main political opponent, called for calm and an end to violence on Wednesday.
"No destruction, revenge or vengeance," BNP leader Khaleda Zia, 78, said in a video address from her hospital bed to hundreds of her supporters at a rally in Dhaka.
A policeman aims his weapon at protesters in Dhaka, Bangladesh. (AP PHOTO)
Zia, who was released from house arrest on Tuesday, and her exiled son Tarique Rahman, addressed the rally and called for national elections to be held within three months.
On Wednesday, a court overturned Yunus' conviction in a labour case in which he was handed a six-month jail sentence in January. Yunus had called his prosecution political, part of a campaign by Hasina to quash dissent.
"Let us make the best use of our new victory," Yunus said. (Reporting by Ruma Paul; Writing by ; Editing by YP Rajesh and Richard Chang)
