The Sinn Féin president has said her phone call with Breege Quinn, mother of murder victim Paul Quinn, went "very well".
The two women spoke by phone for around 15 minutes on Thursday evening, and Mary Lou McDonald said she hopes to meet with Mrs Quinn face to face soon.
The brutal killing, and Sinn Fein's response to it, has become a major issue in the Republic's general election campaign, with rivals claiming it is proof of the party's continued defence of IRA violence.
"I have conveyed directly to Breege our profound sympathy at the loss of her son, obviously the family are dealing with very significant trauma and in truth will deal with that trauma for the rest of their lives," Ms McDonald said today.
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"I'm very anxious that the police on both sides of the border need to do their job and advance this investigation.
@qnewsdesk present as @BreegeQuinn speaks to @sinnfeinireland @MaryLouMcDonald about her son's 2007 murder. It followed a row with an IRA man's son. Mrs. Quinn wants Stormont Finance Minister @conormurphysf to say "Paul Quinn is not a criminal" & go to police with information pic.twitter.com/7nPdIAraHI
— Q Radio News (@qnewsdesk) February 7, 2020
"I think it is worthwhile for us to meet, I said that to Breege and I think it's important that the Quinn family and Conor (Murphy) speak to each other.
"Above all Paul had a very very vicious and brutal death and the idea that those who are responsible for that are not brought to justice is, for me, unthinkable."
Finance Minister Conor Murphy apologised on Wednesday for comments made 13 years ago in the wake of Mr Quinn's murder when he branded the South Armagh man a smuggler and criminal.
Mr Murphy's apology represented a dramatic change in position as prior to that he had denied even making the comments about Mr Quinn.
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But Mrs Quinn said she was left disappointed by the phone call on Thursday, claiming Ms McDonald had not committed to telling Mr Murphy to publicly state her son was not a criminal.
The Quinn family have since asked that a further statement from Mr Murphy be issued that Mr Quinn was not involved in any criminality before his death.
Mrs Quinn told the BBC's Stephen Nolan Show: "She said that Conor Murphy's apology was sincere, I said yes, but there is a couple of words he hasn't said and he needs to say them publicly to give us peace of mind.
"She thought the best idea was to sit around a table and talk about it, I said I wouldn't be sitting around a table until Conor Murphy has said those words.
"She said Conor Murphy has written to us, I said we didn't have a letter yet, and I still insisted I wouldn't be meeting with him until he said that Paul wasn't a criminal. But she would not say she would ask him to say that, no matter how often I asked her.
"I said Mary, you are the leader of Sinn Féin, you're a mother like myself, and if your son was taken and beaten and his name blackened, wouldn't you fight teeth and nail to get it cleared? She said she sure would. I said, Mary well then will you go to Conor Murphy and ask him to come out publicly and say those few words? But she still wouldn't say she would say it.
"But we will wait for this letter, I will reply to it, I will keep asking Conor Murphy."
Mr Quinn, a 21-year-old from Cullyhanna in south Armagh, was beaten to death by a gang of around a dozen men in a farm shed across the border near Castleblayney, Co Monaghan.
His family blame members of the IRA, but Sinn Féin has long denied republican involvement.
A month after the murder, Mr Murphy, a Sinn Féin representative for Newry and Armagh, claimed Mr Quinn was involved in "smuggling and criminality".
While Mr Quinn's family and other political representatives have demanded his resignation, DUP leader Arlene Foster declined to add her voice to those calls when she appeared before an Assembly committee on Wednesday.
Ms McDonald has insisted Mr Murphy will be staying in post.
