Bevins, a co-sponsor of the bill, said she supports the legislation but is concerned about language that many fear opens the door to allowing men, who now identify themselves as women, to use women's bathrooms.
"It's an issue for (constituents) so it's an issue for me," said Bevins.
On the whole, Bevins said she continues to support the bill.
"I'm against discrimination," said Bevins. "I don't think anyone should be discriminated against."
Other council members said the language regarding public accommodations has drawn the most opposition.
Quirk, in previous interviews, said the law does not require businesses to allow transgender people to use the bathroom of their choice.
There now appear to be enough members of the council to approve an amendment that would eliminate the language from the proposed law.
Quirk has declined to say if he would be open to amending out the language, which has been a sticking point in similar legislation at the state level.