The union, Utility Workers Union of America Local 1-2, was "perusing" the utility's "best offer," said union spokesman Joe Flaherty.
"There's myriad possibilities," Flaherty said.
The union had said its 9,000 members might walk off the job at midnight if negotiations didn't result in a better contract offer.
Most of the workers maintain the utility's gas, electric and steam delivery systems.
Con Edison spokesman Michael Clendenin, who confirmed negotiations were continuing, has declined to characterize the tenor of the talks. He has said the giant utility remained hopeful a deal would be reached. A strike wouldn't disrupt service, he said.
Clendenin said Con Edison managers about half of whom rose through the utility's ranks would respond to any emergencies if workers struck. Con Edison has nearly 14,000 employees in all.
Non-emergency repairs and meter reading could be delayed, Clendenin said.
"We are hopeful that we'll get an agreed-upon contract," he said. "But we're preparing for anything."
The negotiations were taking place at a hotel in East Rutherford, N.J. Flaherty said the two sides remained at odds the previous evening over issues including wages, health care costs and pensions.
"They haven't put anything reasonable on the table yet," Flaherty said. He said the union hoped to avoid a strike, but "it really, really doesn't look good."
The union's membership has authorized a strike if a deal isn't reached, but Flaherty said a walkout would not be automatic if the midnight deadline comes without a deal. Union leaders could decide to keep talking or to bring the utility's last offer to members for a vote, among other possibilities.
Con Edison provides electricity in most of New York City and Westchester County and supplies natural gas service in much of the same area. The utility also operates a massive steam system that heats and cools thousands of buildings.
Negotiations between the union and the utility have gone down to the last minute several times before. The last time it led to a strike was in 1983.