NEW YORK (AP) – Mayor Michael Bloomberg and his police commissioner insisted Friday that they did the right thing by going public with a terrorist threat to bomb the New York subway, in the face of suggestions from Washington that they overreacted to dubious information.

Thousands of extra officers flooded the city’s subway system, pulling commuters out of rush-hour crowds and rifling through their bags or briefcases. And in Iraq, authorities detained a third suspect in the plot and investigated whether a fourth had traveled to New York as part of the scheme, according to a law-enforcement official familiar with the case.

The official said the man’s trip to New York was described by an informant who had spent time in Afghanistan and proved reliable in past investigations. But the official, who spoke on condition of anonymity because of the ongoing investigation, added that authorities had not confirmed whether the fourth man even exists.

Alarmed by the informant’s report of a plot to attack city subways with bombs hidden in bags and possibly baby strollers, U.S. forces in Iraq arrested two suspected plotters who had been under close surveillance until Thursday morning, officials said. The third escaped but was captured Friday.

A Department of Homeland Security memo obtained by The Associated Press said the attack was reportedly scheduled to take place on or about Oct. 9, with terrorists using timed or remote-controlled explosives hidden in briefcases, suitcases or in or under strollers.

The memo said that the department had received information indicating the attack might be carried out by “a team of terrorist operatives, some of whom may travel or who may be in the New York City area.”

The memo, issued Wednesday to state and local officials, said that homeland security and FBI agents doubted the credibility of the information, but it provided four pages of advice about averting a possible attack, including tips on inspecting baby carriages with bomb-sniffing dogs.

The announcement Thursday sparked behind-the-scenes jostling with homeland security officials in Washington, who downplayed the threat and said it was of “doubtful credibility.”

But Bloomberg and Police Commissioner Ray Kelly vigorously defended their decision.

“We did exactly the right thing,” Kelly said Friday.

The mayor’s chief spokesman said that the city had been working closely with the FBI, whose New York office head appeared with Bloomberg and Kelly at a Thursday press conference announcing heightened security.

Spokesman Ed Skyler characterized disagreement over the plot intelligence as a dispute inside the federal government.

“When different federal agencies have different interpretations of the intelligence, the mayor doesn’t have the luxury of knowing which one is right,” Skyler said. “He has to make a decision that provides the maximum level of protection to the people of New York.”

Kelly, Bloomberg and other city officials declined to release details of the alleged plot, saying much of the information was classified. They stressed that they had made the plot public because it was in the best interests of the city.

President Bush, asked Friday if he thought New York officials had overreacted, said: “I think they took the information we gave and made the judgments they thought were necessary.”

The dispute played out as new details emerged about the alleged plotters.

Those arrested had received explosives training in Afghanistan, the law-enforcement official said Friday. They had planned to travel through Syria to New York, and then meet with operatives to carry out the bombings.

A federal official said one of the suspects arrested in Iraq apparently told interrogators that more than a dozen people were involved in the plot, and that they were of various nationalities, including Afghans, Syrians and Iraqis. The official spoke on condition of anonymity because the investigation is ongoing.

Several of these details related to the suspects were first reported by ABC News.

In Baghdad, spokespersons for the U.S. military and the U.S. Embassy declined to comment about the arrests. Department of Homeland Security spokesman Brian Doyle also said the government has no information that the fourth person possibly connected to the plot “is either here or even exists.”

Pentagon spokesman Bryan Whitman said the military obtained intelligence information “during the normal course of our operations.” The intelligence led to a military raid in Iraq this week that was conducted by Iraqi security forces, backed by U.S. troops, Whitman said, but added that there was no direct link between the raid and the New York subway threat.

On the streets of New York, more officers were visible in several parts of the city, and one lane of traffic on Ninth Avenue was reserved for emergency vehicles.

“Hopefully, God’s with me and I’ll be OK,” Vinnie Stella said while clutching newspapers under his arm as he entered the subway at Penn Station.

Authorities briefly closed part of Penn Station earlier Friday after a discarded soda bottle filled with a green liquid was found during morning rush hour. Kelly said it was a type of drain-cleaner substance and not a threat.

The boost in police presence in the subways led the New York Civil Liberties Union to ask for a 60-day delay in a federal case brought in Manhattan challenging police searches of bags as unconstitutional because they are discriminatory and ineffective.

The city responded by asking that the case be thrown out and the judge questioned what was left to litigate, setting a hearing for next week.

An estimated 4.5 million passengers ride the New York subway on an average weekday. The system has more than 468 subway stations. In July, the city began random subway searches in the wake of the train bombings in London.