Mayor Bloomberg said Dzhokhar Tsarnaev, the surviving suspect, had told the FBI he and his brother “spontaneously” decided that New York would be next.
Police Commissioner Raymond Kelly told reporters the suspects had a pressure cooker bomb and five pipe bombs.
Three people died and over 260 were wounded in the 15 April Boston attack.
“Last night we were informed by the FBI that the surviving attacker revealed that New York City was next on their list of targets,” Mr Bloomberg said during Thursday’s news conference at city hall.
“He and his older brother intended to drive to New York and detonate those explosives in Times Square.”
Hijacked driver’s escape
Mr Kelly said the brothers had planned to head to New York after hijacking a car and its driver in Boston last Thursday night.
“That plan, however, fell apart when they realised that the vehicle they hijacked was low on gas and ordered the driver to stop at a nearby gas station,” Mr Kelly said, adding that the driver escaped and alerted police.
Police intercepted the brothers in the stolen car, prompting a gun battle that left Tamerlan Tsarnaev dead.
Mr Bloomberg praised Massachusetts law enforcement for their work in stopping the suspects, saying “we know they had the capacity to carry out the attacks”.
The younger brother, Dzhokhar Tsarnaev, is said to have travelled to New York at least once last autumn.
He is now in police custody in hospital and has been charged with using a weapon of mass destruction to kill people.
Before reportedly telling investigators he and his brother planned an attack on Times Square, he had said they were planning to go to New York “to party” after the bombings.
Following 16 hours of interrogation, Mr Tsarnaev stopped responding to investigators’ questions after being read his legal rights to remain silent and have a lawyer, US media report.
Mr Bloomberg said there was no evidence New York was currently a target, but that the Tsarnaevs’ alleged plan proved the city remained a prime location for people who want to “bomb and kill Americans”.
On Thursday afternoon, police vehicles lined Times Square in a show of force, with officers standing shoulder to shoulder.
“Why are they standing like that? This is supposed to make me feel safer?” Elisabeth Bennecib, a tourist from France, told the Associated Press.
“It makes me feel more anxious, like something bad is about to happen.”
Terror watch list
The suspects’ father, Anzor Tsarnaev, has said he will travel imminently to the US. The family wants to take the dead son’s body back to Russia. But it is not clear if his former wife will join him.
Zubeidat Tsarnaeva left the US and failed to make a court appearance after being arrested last June on suspicion of stealing $1,624 (£1,050) of women’s dresses from a Massachusetts department store.
At a news conference on Thursday in the Russian republic of Dagestan, where she now lives, she angrily denied her sons had anything to do with the bombings.
Meanwhile, there are questions as to whether the authorities did enough to prevent the bombings.
Russian officials are understood to have contacted the FBI in 2011 on suspicion Tamerlan Tsarnaev had become a follower of radical Islam, but investigators found nothing and closed the case.
He was added to a terrorism watch list 18 months ago at the CIA’s request, according to US media reports.
US officials said earlier their intelligence community had no information about threats to the marathon ahead of last week’s attacks.