“I would certainly consider that,” he said at the beginning of a cabinet meeting. “Send him to Gitmo, I would certainly consider that, yes.”

Likewise, he vowed to toughen prosecution and punishment of terrorists without specifying how. “We need quick justice and we need strong justice, much quicker and much stronger than we have right now,” Mr. Trump said. “Because what we have right now is a joke and it’s a laughingstock.”

His later call for the death penalty stemmed from news that Mr. Saipov expressed loyalty to the Islamic State even as he recovered from a gunshot wound by a police officer who stopped his rampage. In a criminal complaint filed on Wednesday, the F.B.I. said Mr. Saipov “requested to display ISIS’s flag in his hospital room and stated that he felt good about what he had done” during an interview with law enforcement officers.

No one arrested on American soil has ever been sent to Guantánamo Bay, and no one captured on foreign soil has been sent there since 2008. Transferring the suspect from New York would raise a host of thorny constitutional and legal issues, and Mr. Trump seemed to be speaking off the top of his head since federal prosecutors later moved to process the suspect in civilian courts.

Asked later about his comment on Guantánamo, Sarah Huckabee Sanders, the White House press secretary, dismissed it as notional, saying that “he wasn’t necessarily advocating for it, but he certainly would support it if he felt like that was the best move.”

By the end of the day, the president came under criticism from conservative allies for not declaring Mr. Saipov an enemy combatant, which would have allowed interrogators more freedom to question him without granting him the rights of a civilian defendant. “The Trump administration missed an important opportunity to send a strong message to terrorists and make America safer,” said Senator Lindsey Graham, Republican of South Carolina. “This is a huge mistake. Very sad.”

The diversity visa program cited by Mr. Trump was created in 1990 by a bill supported by Mr. Schumer, passed by bipartisan votes and signed into law by a Republican president, George Bush. Mr. Schumer supported getting rid of the program as part of a comprehensive immigration plan crafted by eight lawmakers and passed by the Senate in 2013. But the plan was blocked in the House by Republicans who objected to other elements they considered amnesty for illegal immigrants.