A police report said Mohammed Hassan Cheikh Mohamud collided with the courier in broad daylight in Istanbul.
Police say he was driving a car which reportedly belonged to the Somali consulate there.
Turkey imposed a travel ban on Mr Mohamud but officials believe he has already fled the country.
In a post on X, formerly known as Twitter, Istanbul's mayor Ekrem İmamoğlu shared grainy CCTV footage of the 30 November incident.
A car can be seen appearing to hit the back of a motorcycle at a junction on a busy multi-lane road.
The motorcyclist, father-of-two Yunus Emre Göçer, 38, was taken to hospital but died from his injuries six days later on Wednesday.
An initial police report had said that Mr Göçer had been at fault and therefore Mr Mohamud was released, according to Mr Göçer's lawyer quoted by Turkish media outlets.
But a review of the security footage suggested there may have been another explanation.
Mr Mohamud, who is the son of Somali President Hassan Sheikh Mohamud, has not commented.
An international arrest warrant for Mr Mohamud was issued by the Turkish prosecutor on Friday.
This was after police had visited his home to find that he had not been there for almost a week, the AFP news agency says, citing local reports.
Turkey and Somalia have enjoyed growing diplomatic relations over the past decade, with Turkey providing infrastructure investment, military training and humanitarian aid.
BBC