Mayor Marty Walsh is still hopeful that a deal allowing gay groups to march in the South Boston St. Patrick’s Day parade on March 16 will happen even as the parade organizers and one of the groups appear to be far apart.

Parade organizers and MassEquality, an advocacy group that played a crucial role during the state’s gay marriage debate over a decade ago, met behind closed doors on Monday with Walsh and Congressman Stephen Lynch in attempts to hammer out a compromise. But based on the latest round of statements that both camps issued to reporters, a deal looks out of reach at the moment.

Walsh has marched in the parade as a Dorchester state representative, but during the mayoral campaign last year, he pledged he wouldn’t march if parade organizers continued to exclude gay groups. After the meeting, Walsh noted that the Dorchester Day Parade, which features a DotOUT float and is held every June, doesn’t have the issues as the St. Patrick’s Day parade. “There’s never been an issue around who can march, who can’t march,” he said. “And hopefully we can get to the same point with the St. Patrick’s Day parade.”

Walsh said the two camps have miscommunicated with each other while putting out dueling statements and talking to reporters. “I think part of the problem is, yeah, reporting put out there, rather than letting some discussions happen,” he said. “I think if we could have discussions without – I’m not blaming the media for it.

Misinterpretation of certain things, I think, has happened here. So hopefully we can get this back on track.” Walsh, an Irish-American Catholic who backed gay rights early in his political career, hosted the two camps in his office earlier this week for talks.

In 1995, the US Supreme Court ruled that parade organizers were exercising their First Amendment rights in choosing who could and who couldn’t participate in the parade. But the latest release from the parade organizers, a typo-laden screed, came on Tuesday, laying out their reasoning for the rejection of gay or LGBT (lesbian, gay, bisexual, and transgender) groups.

An affiliate of MassEquality, LGBT Veterans for Equality, had applied to march in the parade. The Allied War Veterans Council, one of the organizing groups, rejected it, saying it was a “clear violation of our ‘No sexual orientation’ rule, and not that we ban Gay people as reported by the press,” the statement said.

The application, which was later reconsidered by parade organizers and rejected again this week, said 20 veterans would march in the groups. The parade organizers gave conditional approval, saying the veterans could march “but no sexual orientation would be displayed. Kara Coredini, the executive director of MassEquality, rejected the offer, saying they would only accept if they were allowed to “march openly and honestly.”

“Her rejection was based on the fact that we would not allow LGBT Veterans to identify themselves as openly Gay by means of signage and T Shirts Identifying Them as LGBT Veteran,” parade organizers wrote in their statement. “This clearly violates our code of conduct.”

The organizers added that at the meeting in the mayor’s office on Monday night, the LGBT veterans group confirmed it did not have 20 veterans available who would march with them. “We appreciate the efforts the Mayor’s office to try and mediate this issue,” the organizers said. “However it is our intention to keep this an Irish Celebration, dedicated to our Men and Women serving in our Armed Forces. We will fight to keep our parade and its traditions.”

Earlier, MassEquality had put out a statement that struck a more hopeful note, in line with what the mayor had been pushing. “No agreement was reached, but the conversation is ongoing,” Coredini said. “We hope that the parade organizers will see exclusion as not only unfair, but harmful and not aligned with the many fair-minded and inclusive Bostonians – Irish or not – who participate in and enjoy this annual celebration of a community that is so vital to the city of Boston.”