The allegations were made in a BBC Spotlight programme on Tuesday.
Mr Adams has denied any involvement  in the murder of Mr Donaldson in 2006.
In a statement, BBC Spotlight said: "The Spotlight programme dealt with matters of great public interest and the BBC stands by its journalism."
A lawyer for the Donaldson family spoke to the BBC on Friday after meeting Garda Deputy Assistant Commissioner Derek Byrne and two other senior police officers.
Ciaran Shiels said the officers provided an update on progress in their investigation during the meeting in Belfast.
"The one theme that has come out of today's meeting is that the theory that was being advanced by BBC Spotlight earlier this week, that this was in some way carried out by the Provisional IRA or authorised by Gerry Adams, I think it's absolute nonsense," he said.
Asked how the family responded to the claims that Mr Donaldson was killed by the Provisional IRA, and that the killing had been sanctioned by the Sinn Féin leader, Mr Shiels said: "It just does not stand up.
"It does not marry in any way with the lines of inquiry that have been progressed by the guards ([Garda] or by the the (police) ombudsman."
A man who said he worked as an informer for the intelligence branch of the police told Spotlight that he had been told the Provisional IRA had killed him.
The former spy, who was in the IRA and Sinn Féin., told the programme that in his experience such a murder would have to have been sanctioned by Gerry Adams.
The man could  not 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.
Mr Donaldson was shot dead in a remote cottage in Donegal in April 2006, months after admitting that he had been an MI5 agent for more than 20 years.
Three years later, the Real IRA said it was responsible.
Mr Donaldson had worked for Sinn Féin as an administrator at Stormont.

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The midfielder, 28, is under contract with Arsenal until 2018 but speculation over his future has been widespread.
"It is definitely my preference to stay," the German said on Wednesday.
"Once everyone is back in London, we will sit down and discuss the future. I have always said that I feel very good at Arsenal."
Arsenal face Sydney FC on Thursday (11:00 BST) and Western Sydney Wanderers (11:00 BST) on Saturday before heading to China for two further games.
Arsene Wenger's side meet Bayern Munich (12:20 BST) on 19 July in Shanghai and Chelsea (12:40 BST) on 22 July in Beijing before returning to England.
"For now the most important thing is our pre-season and getting through the tour, training and getting fitness," added Ozil, who joined Arsenal from Real Madrid for £42.4m in 2013.
"I have always said that I feel very good at Arsenal."
While Ozil appears keen to stay, the futures of Chile forward Alexis Sanchez and France striker Olivier Giroud remain less clear.
Giroud, 30, has been with Arsenal for five years but has been linked with moves to Premier League rivals Everton and West Ham.
"I've been through some difficulty this last year, but always succeed to bounce back," said the former Montpellier forward.
"I'm still an Arsenal player and try to be professional like I always have done and to prepare well for the next season."
Meanwhile, Sanchez has been linked with Manchester City.
"It would hit the team quite hard if he left because he is a player who always delivers," said Ozil.
The family of murdered Sinn Féin official Denis Donaldson have said they do not believe the Provisional IRA killed him, or that his shooting was authorised by Gerry Adams.
[NEXT_CONCEPT]
Mesut Ozil says he wants to stay at Arsenal and will discuss his future with the club after a pre-season tour of Australia and China.