Broadway shows continued to perform Wednesday even as the Rialto teetered on the verge of a shutdown, with contract talks between producers and the stagehands union in a deep-freeze impasse.

The League of American Theaters and Producers publicly rejected on Wednesday the last, best offer laid on the table Tuesday night by the union, Local One.

Producers, indicating they remain open to a “serious response” from the union, gave no indication of any potential timeline for a management-initiated lockout of Local One that would darken the majority of Broadway shows.

The lockout remains the most urgent possibility since a strike could not occur before a special meeting of union membership is held to vote on approving the labor action — a vote that would be followed by a request for strike authorization from the international leadership of umbrella union IATSE. Local One has planned a Saturday meeting for its New York members, but this session is not designated a strike vote, according to a rep.

Some in the industry have begun to hear whispers that IATSE international prexy Thomas C. Short has privately told Broadway theater owners that he would not authorize a strike should the local, headed by James J. Claffey Jr., request it.

Both the international and the local office had no comment, but the rumors seemed plausible thanks to long-standing (and far from secret) friction between the local and international leadership of the stagehands union. During the 2003 Broadway musicians strike, Short reportedly assured producers that Local One would not honor the musicians’ picket lines — but union members did so anyway.

If the current chatter is true, the move could seriously undermine the local’s bargaining power with producers. One theory holds that producers will forgo a lockout, instead simply refusing to budge on contract terms because they know the local will not be authorized to strike.

Neither side, however, would acknowledge such talk on the record.

In rejecting the union’s final offer, league exec director Charlotte St. Martin said in a statement, “The union’s proposal insists on retaining egregious contract features.”

She pointed specifically to the offer’s requirement of a flyman on all productions, even on shows with no fly cues. League estimated an unnecessary fly operator would cost producers more than $160,000 a year.

Local One, for its part, refused to discuss specific negotiation points in the press. The union’s last, best offer was issued late Tuesday night in response to the final offer the producers had laid out earlier that evening.

Tuesday’s showdown came after the League and Local One had been locked in contentious contract renegotiations for several weeks.

The central sticking points are work rules governing the number of stagehands whom producers are contractually required to employ, particularly for the process of installing a physical production into a Broadway theater.

Producers want to achieve flexibility in rules they consider outmoded and indefensibly expensive, while stagehands contend that such obligations are protections of their livelihood that they will not give up without an equal exchange.

Stagehands have been working without a contract since the previous agreement expired at the end of July.