He made the allegation to BBC Northern Ireland's Spotlight programme on Tuesday night.
Mr Donaldson was shot dead months after admitting in 2005 he had been an MI5 agent for more than 20 years.
Mr Adams has denied any involvement in the killing of Mr Donaldson.
He told radio station LMFM that he "totally and specifically refuted the allegation I was consulted".
Mr Donaldson had worked for Sinn Féin as an administrator at Stormont.  He was killed at a remote Donegal cottage in 2006.
A man who says he worked as an informer for the intelligence branch of the police told Spotlight that Mr Adams sanctioned the murder.
The former spy was in the IRA and Sinn Féin. He cannot be identified because of fears about his safety.
What Spotlight was told
Agent: I know from my experience in the IRA that murders have to be approved by the leadership. They have to be given approval by the leadership of the IRA and the military leadership of the IRA.
Presenter: Who are you specifically referring to?
Agent: Gerry Adams, he gives the final say.
The informer told Spotlight that murders had to be approved by the political and military leadership of the IRA.
In a statement, Mr Adams' solicitor said the Sinn Féin leader "has no knowledge of and had no involvement whatsoever in the killing of Denis Donaldson".
He added that Mr Adams "categorically denies the unsubstantiated allegation that he was consulted about an alleged IRA army council decision or that he had the final say on what had been sanctioned".
Days after the murder, the IRA said it was not involved in Mr Donaldson's death.
But, security sources have told Spotlight that intelligence received following the killing contradicted the IRA's denial.
You can watch BBC Spotlight: Spy in the IRA now on BBC iPlayer.
A man who says he was a former IRA and Sinn Féin member turned British agent has claimed Sinn Féin leader Gerry Adams sanctioned the killing of Denis Donaldson.