Al Jazeera's Monica Villamizar, reporting from Kingston, the Jamaican capital, said that the minister of justice had given the final sign-off on the extradition.
"There are a lot of expectations here, because this man has the potential to stabilise the whole country," she said.
Coke, who is accused of leading the "Shower Posse" gang, was arrested by Jamaican police on Tuesday after eluding a bloody offensive in his neighbourhood.
The assault by police and soldiers in the Tivoli Gardens slum in Kingston lasted four days and 76 people, mostly civilians, were killed.
Coke, 42, is wanted in the US on drug-trafficking and gun-running charges.
"We look forward to working closely with the Jamaican authorities to bring Coke to justice to face charges pending against him in the United States," Rebecca Park, a spokeswoman for the US embassy, said Wednesday.
Police arrested Coke at a road checkpoint in the Portmore area of St Catherine Parish on the outskirts of Kingston on Tuesday.
But Miller, an influential evangelical preacher who facilitated the surrender of Coke's brother earlier this month, said that Coke was going to surrender at the US embassy when police stopped his vehicle.
"A contact was made on his behalf that he wanted to give himself in," Miller said.
"I therefore made arrangements with his lawyers because he wanted to go ahead with the extradition process, so we communicated with the US embassy because that's where he would feel more comfortable."
Miller, a minister at the nondenominational Whole Life Ministry, said after capturing Coke, police took him to the nearby Spanish Town police headquarters.
He said Coke contacted him to ask his help in arranging the surrender at the embassy because he did not trust the police not to harm him if he surrendered to them.
"He also wanted to waive his right to an extradition hearing so that he could go to the US for a trial," Miller said.
Coke's father died in a mysterious prison fire while he was awaiting extradition.
Coke was known as "president" to the people of the Tivoli Gardens slum, and served as community leader and enforcer in the neighbourhood in an area that the government acknowledges it has long neglected.
"When people go to the police to make complaints, the police just say 'Go home, we'll come see you soon'," Everton Cornpipe Jones, Coke's brother, told Al Jazeera.
"But they never come and then anything can happen to you."
Coke also commanded a private militia and was a strong supporter of the ruling Jamaica Labour party.
US prosecutors have described Coke as the current leader of the Shower Posse gang, which murdered hundreds of people by showering them with bullets during the cocaine wars of the 1980s.
Describing the latest developments, Ross Sheil, a journalist with the Observer newspaper, told Al Jazeera: "This has come at the end of what has been a very turbulent month for Jamaica, so people are generally very relieved."