The accusations are based on the claims of a former council officer.
Gordon Matheson is now the subject of a complaint to Scotland's ethics watchdog alleging he repeatedly violated the Councillors' Code of Conduct during the design competition for the £15 million project.
Matheson last night said he was confident of being cleared.
He faces claims of interfering in a legally binding procurement process, improper bias, attempted staff coercion and of trying to "steer" the contest in favour of his preferred design.
The complaint was sent last week to the Public Standards Commissioner, who has the power to disqualify councillors, by the Royal Incorporation of Architects in Scotland (RIAS), the professional body which ran the £100,000 design competition for the council.
It includes claims from Kerr Robertson, an RIAS member who, until his retirement a few weeks ago, was the council's lead architect and projects director.
Robertson claims that, despite the contest having to comply with EU procurement law, he was told six weeks before the final judging began that "Gordon Matheson [would] be allowed to choose the winning design".
Robertson also claims he was told "to ensure the other jury members would fall into line with this", but he objected.
The SNP said Matheson's position was "increasingly untenable".
A council source dismissed the complain as "pretty weak stuff" and suggested that Robertson may have a grudge against Matheson.
Labour's 2012 manifesto pledge to revamp George Square ended in farce in January after the judging panel picked a winner from the six entries shortlisted, only for Matheson to announce within minutes that the council had axed the plan.
The Sunday Herald later reported claims that Matheson had wanted a design by architects Burns + Nice to win, but his fellow judges instead chose a design he vehemently disliked, by John McAslan & Partners. Matheson's preferred option came fourth.
Although Matheson blamed the U-turn on public opposition, council sources claimed he had "thrown his toys out the pram" after his favourite was rejected.
The square is now getting a modest "facelift" rather than the planned revamp. The spending watchdog Audit Scotland is already looking at the fiasco.
The RIAS complaint takes the issue to a far higher level and increases pressure on Matheson ahead of the Glasgow Labour Group AGM on May 13, when he could face a challenge to his leadership.
The Standards Commissioner, who investigates complaints against councillors, MSPs and quango members, has the power to suspend or disqualify those found guilty of breaching codes of conducts.
In Matheson's case, he could in theory be disqualified until the next council election in 2017, effectively ending his political career if the complaints were upheld.
The RIAS last month produced a scathing report blaming Matheson for the George Square "debacle", saying it brought Glasgow into "significant disrepute" and cost taxpayers £100,000 and architects £200,000.
Filed at the request of the RIAS ruling council, the follow-up complaint to the Standards Commissioner centres on Matheson's "alleged misconduct".
It says the RIAS has "very significant concerns about the running of the competition and the role of Councillor Matheson in particular".
After listing possible code of conduct breaches over bias, treatment of staff, and interfering in procurement, it concludes: "In all of the above we are of the view that Councillor Matheson is found wanting."
Graeme Hendry, leader of the council's SNP opposition, said: "George Square is the fiasco that simply will not go away for Gordon Matheson.
"For a highly respected organisation such as the RIAS to take the unprecedented action of reporting a council leader to the Standards Commissioner is very damaging for the council and Labour in Glasgow.
"The allegations ... make Councillor Matheson's position increasingly untenable.
"Should he be found guilty of any of these accusations he will be left with no choice but to resign. However, given the high-profile role he is likely to have over the next 18 months he should now ask himself if Glasgow would be better placed with a Leader whose probity is not under challenge.
"Would Glasgow benefit from someone who could give 100% to the significant challenges the city faces and the hosting of the Commonwealth Games as opposed to someone so publicly damaged?"
TORY councillor David Meikle, who secured the Audit Scotland scrutiny of the George Square fiasco, added: "These are serious and disturbing allegations. The fact that a ... professional body both felt obliged to complain is highly significant. It is making Gordon Matheson's position untenable."
An RIAS spokesman said: "We do not comment on leaked documents."
At the Scottish Labour conference in Inverness yesterday, Matheson said he had not seen the complaint but was "entirely confident" of being cleared.
"I'm relaxed about this," he told the Sunday Herald.
He added the SNP was "overplaying its hand" with the call for him to step aside. "They would say that, wouldn't they?" he said.
A council spokesman said: "Even the RIAS make it clear in the complaint that they can't find a rule they think has been broken and they acknowledge that the competition was well run.
"They are also clear that they have no evidence of anyone attempting to improperly influence the jury. What is clear is that there was no public appetite for a radical redesign of George Square and that whichever design won we would have gone with the public's view."