'"He needs to come out and respond to it directly, address the people of Toronto and give his version of the story," Councilor Peter Milczyn said. "Either he's adamant that it's untrue or he has another explanation."
Ford dismissed the video and two published reports as "ridiculous" on Friday.
While he addressed City Council members in chambers Tuesday, Ford's comments didn't include reactions to the purported video, the Canadian Broadcasting Corp. reported.
Instead, Ford spoke during council debate on a casino, which the panel later voted to reject.
The purported video allegedly was seen by Toronto Star reporters and the U.S. gossip website Gawker, the CBC said. Toronto Star and Gawker reports indicated the video shows a man believed to be Ford smoking what appears to be crack cocaine.
"They're disturbing, they're sad, they need to be responded to one way or another," Councilor Mike Del Grande said of the reports.
Councilor Giorgio Mammoliti said if he were the mayor, he would be trying obtain the alleged tape.
"I think he should be asking for the video," Mammoliti said Tuesday. "If somebody's got a video out there, I think the mayor should be saying, 'Hey, where's the video? If you've got something, then show it.'"
Notable deaths of 2014 [PHOTOS]