But the public and media outcry over the attack has fueled concerns that people who are dangerously unstable are wandering the streets, and the random nature of the assault reminds New Yorkers of the schizophrenic who pushed a woman into the path of a subway train last January.

Of the attack on Ms. Barrett, Mr. Giuliani said on his radio show that ''the presumption is that this was done by a mentally ill person.'' He added that ''you're going to have a lot of fear, you're going to have a lot of reaction, and you're going to have a lot of over-reaction.''

Mr. Safir said the police would probably not be stepping up efforts to roust the homeless from the streets had the brick attack not occurred.

Mr. Giuliani, who is a likely candidate for the United States Senate against Hillary Rodham Clinton, has taken credit for the dramatic drop in crime over the last several years. He is broadcasting television commercials upstate that stress his achievements, including the reduction in crime that he considers a cornerstone of his administration.

Although Mr. Giuliani said on his radio show that he did not agree with the Daily News headlines, he said he did agree with the substance of the editorial, which said that while not all mentally ill people on the streets were dangerous, ''a few of them are murderous.'' The editorial urged the city to ''get them off the street. Now.''

The police began rounding up the homeless in a series of Manhattan sweeps in the mid-1990's, and many either moved out of tourist areas and went to less patrolled places -- or faced arrest and court summonses for such public nuisance violations as impeding the flow of pedestrian traffic. Police officials said yesterday that the homeless could be arrested on charges of disorderly conduct or trespassing.

Advocates for the homeless said yesterday that Mr. Safir's announcement would be the first time that the city had threatened the homeless with arrest as a broad policy. Police officials said it was simply a ''re-emphasizing'' of existing policy.