A counter offer has been made to Metro Transit director Eddie Robar, an Amalgamated Transit Union official confirmed to CBC News.
Details of the offer haven't been released.
Ken Wilson, president of Amalgamated Transit Union local 508, said the offer, which he presented just after noon Monday, protects his members right to select their own shifts.
Wilson said Friday the union wasn't satisfied with the language written in management's most recent offer about rostering-style scheduling.
"Nothing would change the way we pick our work," said Wilson.
"We're not reinventing the wheel. We're just coming up with an out-of-the box idea that will fill that open work without generating overtime and that's what council and ministers have been saying all along. Both sides have missed that message, and we're trying to get back on message."
Officials with Metro Transit are reviewing the offer and will present it to city council Tuesday.
The ATU local 508, with about 750 employees, walked off the job Feb. 2.
Phil Wilson, a commuter who has been walking since the strike began, said his support for the union has wavered.
"In the past I supported the strikers, I really think it's time to seriously get back. It's enough, let's get back to work," he said Monday morning.
The city's offer included a five-year contract with $1,500 signing bonus and 2.25 per cent increase in each of the last four years.
Union member Adam Francis said the city's offer looked like an attempt to buy support, because he said the main issue remains control over bus driver schedules.
"It isn't about the money, that's what they do is try to buy our vote — it's rostering," Francis said. "Until the city changes its mind, it's going to be a fight."
"Give and take a little bit on the negotiations, that's what negotiations are," Phil Wilson said. "I'm disappointed. I really thought for Ken Wilson to take it back to the members, it must've been something substantial."
Meanwhile, an online petition began Saturday. A group saying it represents citizens of HRM is calling for arbitration.
The Facebook petition had more than 220 signatures as of Monday morning.
With Tuesday's forecast calling for snow, Phil Wilson said he's just hopeful he can be back on the bus soon.