Government ministers are furious about photographs of the covert training session published by the Herald yesterday.
However, they have concluded the breaches of Covid-19 rules should not lead to a review of their decision to go against health advice in order to allow the return of 'elite' sports later this month.
They hope the GAA will move swiftly to sanction the six-in-a-row All-Ireland champions and continue with planning for the national leagues to go ahead behind closed doors in May.
Sports Minister Jack Chambers said changing the timeline on the basis of what occurred in Dublin would amount to punishing "31 other countries who are complying with public health guidance".
"What has happened in last 24 hours undermines the broader public health message," he said, but added that he expects the GAA to "deal with it seriously and properly".
Officials from the minister's department were in contact with Croke Park early yesterday to express "surprise and disappointment" with the revelations in this newspaper.
Gardaí are also now "making enquires" into the training session which took place at Innisfails GAA club in north Dublin in the early hours of Wednesday morning.
It is understood they are investigating who organised the 'dawn run' and whether any of those in attendance travelled more than 5km to take part.
"If gardaí identify potential breaches of the public health regulations (travel restrictions), a Fixed Payment Notice may be issued where appropriate," a spokesperson said.
The session, involving at least nine players, began barely 12 hours after Croke Park issued a circular to all clubs and counties, warning that any breaches of the current ban on collective training could put the GAA's overall plans for a return "in serious jeopardy".
The group were observed engaged in timed fitness drills while under the instruction of a coach.
Following an internal investigation, Dublin GAA has accepted there was a breach of Covid-19 guidelines and suspended its football manager Dessie Farrell for 12 weeks.
"The Dublin senior football management and players recognise that this was a serious error of judgment and apologise unreservedly for their actions," the county's management committee said.
But GAA headquarters is to carry out its own probe after expressing "frustration and extreme disappointment" at the images which included some of the sport's biggest stars. Among those present were reigning Footballer of the Year Brian Fenton, and two-time All Stars Jonny Cooper and Brian Howard.
In a strongly worded statement, the GAA said it will "invoke any necessary disciplinary processes as appropriate".
The National Public Health Emergency Team (Nphet) made no reference to GAA activities in its most recent advice to the Government.
Instead it suggested that only non-contact sports for children should be allowed to resume in May under strict protocols.
However, sources familiar with the discussions said ministers were keen to get GAA for senior men and women back in time to allow for a full season.
It also granted exemptions for a number of "elite athletes" in other sports.
A source said that ministers believed the GAA has a "significant role" in Irish society and the potential for having games on television during times of restricted movement "would give people a lift".
It was noted that there were a number of Covid outbreaks linked to the aftermath of club finals last year but the intercounty championship had passed largely without incident.
"It was felt that at senior county level they could do it safely again this time. The GAA has already proven they can plan for this," the source said.
GAA officials are expected to announce a new schedule for the season next week, with a possibility that the leagues will commence on the weekend of May 15/16.
No firm decision on the competition structure for the season has been taken yet, with a new, much-shortened timeframe likely to be involved.
Mr Chambers said last year's championship was run in a "smooth and efficient way".
"There was protocols around public health and management of Covid," he said.