The poorly framed video of the largely incoherent Ford using Jamaican swear words was posted to YouTube anonymously, and appears to have been filmed at a restaurant called Steak Queen in the city's north west.
Ford admitted he had been drinking, but said it was on his "own time."
"I was with some friends and what I do in my personal life with my personal friends, that's up to me," Ford told reporters outside his city hall office. "It really has nothing to do with you guys."
The mayor denied that he was on drugs the previous night but wouldn't say how he got home from the restaurant in that state.
In the video Ford uses profanity, appears to imitate a Jamaican accent and uses the Jamaican swear word bumbaclot, references Toronto Police Chief Bill Blair and counter-surveillance, though full sentences cannot be made out from the audio.
Ford said he did not think the language he used was offensive or discriminatory.
"It's how I speak with some of my friends," he said.
A second video was posted to YouTube on Tuesday night purporting to show Ford sitting with a friend at the same restaurant on Monday night.
"When I saw him he was not drunk and he was not even drinking, he was sitting with a friend and respectable," the text with the posted video said. The video does not show the face of the man said to be Ford.
The other man in the video appears to be Ford's friend Alexander Lisi, who police charged last year with extortion, alleging he tried to get his hands on the infamous video that appears to show Ford smoking crack cocaine.
Lisi was freed on bail in November on conditions that include abstaining from possession or use of non-medically prescribed drugs, and not associating with anyone he knows to have a criminal record.
A spokesman for Ford has not responded to an email asking him to comment on whether in fact the second video shows Ford with Lisi.
Coun. Michael Thompson, the city's only black councillor, said the mayor's interpretation of "supposedly being Jamaican" was offensive.
He called it another unfortunate situation in the "unravelling of Rob Ford," saying many fellow councillors were skeptical of Ford's claims of sobriety.
"We've been fairly silent in just waiting for the next thing to occur," Thompson said. "I think we all had sort of concluded it was only a matter of time."
Ford has repeatedly said he hasn't had a drink since November — after a month in which reams of allegations in police documents that he did drugs, drove drunk and consorted with gang members were released by the courts.
The mayor has denied those allegations, which have not been proven in court, but did admit in November that he had smoked crack cocaine, likely in one of his "drunken stupors."
Ford's brother at first denied the video could have been filmed Monday night, both because the mayor looks thinner these days and because he spoke to him at 10:30 p.m.
"Obviously that's him but it didn't happen last night," Coun. Doug Ford told reporters after watching the video. "I'll repeat what he's said 10 million times: Beginning of November he hasn't taken a drink. Very simple."
A user of the social media site Instagram posted a photo around midnight of the mayor posing with two men and identified the location as Steak Queen.
Coun. Joe Mihevc said it appeared from the video that Ford was "off the wagon."
"It was his commitment that he would clean up his act," Mihevc said. "What this shows is that he has not done that."Suggest a correction