Hearts have written to the Scottish FA and SPFL questioning the "troubling precedent" set by the "premature end" to last Saturday's Scottish Premiership title decider at Celtic Park.
The Jambos went into the final day of the season one point ahead of the Hoops, but Callum Osmand's late goal saw a number of home fans run on to the pitch to celebrate.
Hearts players left the stadium immediately while still in their kits due to what the Tynecastle club say was a "menacing and threatening atmosphere inside the stadium".
Celtic apologised to Hearts for the incident, adding they would "co-operate fully with any investigation".
But the Edinburgh side have released a new statement, raising concerns about the season finale.
"As previously noted, the club is investigating events before, during and after the match, and will offer Police Scotland our full support as they continue with their enquiries," the statement read.
"We have also written to the SFA and SPFL, setting out our observations and questions regarding the circumstances surrounding the premature ending of the match, and expressing our concern that a troubling precedent has been set whereby a pitch invasion can effectively determine the duration of a football match, rather than the match officials.
"We have asked them for a prompt response to the points we have raised.
"The incursion of spectators onto the field of play must never be condoned under any circumstances, and the seriousness of the potential consequences must be fully understood by supporters and clubs alike.
Hearts have also hit out at Martin O'Neill's comments on Monday, where he refused to condemn the scenes after Osmand's goal.
The Tynecastle Park club had revealed players and staff reported "serious physical and verbal abuse" during scenes and afterwards, describing them as "unacceptable" and "disgraceful".
However, O'Neill on talkSPORT was asked if the scenes had tainted the image of the game and said: "I'm sorry, I totally disagree with that. I don't know about the confrontations in terms of the Hearts players, and there's a lot of hyperbole about that, let's find out the real picture."
Hearts had added that the aftermath of that third goal had "embarrassed Scottish football". However, O'Neill said: "Well, I don't believe that, I just don't believe it, I think it's nonsense" and continued by adding "it's a home game and we'd just won the league, and the fans have come onto the field, alright? Okay, so they should stay put then?"
In response to those comments, Hearts' latest statement added: "The incursion of spectators onto the field of play must never be condoned under any circumstances, and the seriousness of the potential consequences must be fully understood by supporters and clubs alike.
"We are aware of highly irresponsible comments made publicly this week, which appear to seek to defend those who entered the pitch on Saturday and undermine the gravity of the situation. Those remarks carry dangerous implications, and we call upon the football authorities to ensure that this matter is addressed with the seriousness it demands, and with the full cooperation of all involved."
Your Site News understands the SPFL are set to hold discussions with the Scottish Government over criminalising pitch invasions, as has been the case in England and Wales since 1991. They are also keen to discuss tailgating becoming a criminal offence, which recently became the case in England.
Scotland's First Minister, John Swinney, said such a move would only be considered alongside a consultation on the reform of football banning orders, which closed before the election.
"The Government will engage with the football clubs, with the football authorities and with the local authorities to take what steps we can take to avoid this happening, and if there are further legislative steps that we've got to consider to make sure that these issues are addressed, then we will do that," he said.
Swinney, who was critical of those fans who threw missiles at police officers during celebrations in Glasgow city centre - scenes which resulted in a number of arrests - added: "We've got a job of work to do, working with the football clubs, and the SPFL, and with the City Council to avoid this situation ever happening again. And I commit the Government to engaging in that process.
"But fundamentally, at the heart of this, is the unacceptable behaviour of the minority of fans and that has to be addressed."
