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Trump also said he would ask Congress to terminate the visa lottery program under which the suspect Sayfullo Saipova, a native of Uzbekistan, entered the country.
During a meeting with his Cabinet at the White House on Wednesday, Trump told reporters the US government needed to be much tougher and smarter and less politically correct and accused previous administrations, particularly that of predecessor Barack Obama, of being overly soft on immigration and terrorism, Efe news reported.
"Send him to Gitmo - I would certainly consider that," Trump, using a nickname for Guantanamo, said when asked about that possibility by one reporter.
No one detained on US soil has ever been sent to the Guantanamo Bay detention camp, located on a US military base in Cuba, and no suspected terrorists captured abroad have been transferred there since 2008.
Obama vowed to close Guantanamo upon taking office in early 2009; although he did not keep that promise, he reduced its population from 242 inmates to 41 by transferring nearly 200 individuals to third countries.
While awaiting fresh details about the 29-year-old Uzbek suspect, the Department of Homeland Security confirmed to Efe news on Wednesday that Saipov had entered the country under the so-called Diversity Visa Lottery Program, which offers up to 50,000 visas per year to nations with a low rate of immigration to the US.
Trump had been critical of that program prior to Tuesday's attack, but on Wednesday he went a step further.
"I am today starting the process of terminating the diversity lottery program. I am going to ask Congress to immediately initiate work to get rid of this program," Trump said.
"We want a merit-based program where people come into our country based on merit and we want to get rid of chain migration," Trump added, telling reporters that Saipov had been allowed to bring in 23 other people as a "primary point of contact."
The Diversity Visa Lottery Program originated in a bill that was introduced in 1990 by US lawmaker Chuck Schumer (then a member of the US House of Representatives) and subsequently absorbed into a broader immigration bill that was passed with the votes of Democrats and Republicans and signed by then-President George H.W. Bush.
Investigators say Saipov on Tuesday used a rented Home Depot pickup truck to run over several people on a crowded bike path along the Hudson River in Lower Manhattan, killing eight and injuring 13 before smashing into a school bus.
The suspect, who eyewitnesses say screamed "Allahu Akbar," Arabic for "God is great," and waved a paintball gun and a pellet gun after stepping out of the vehicle, was shot at least once in the abdomen by police and is now hospitalized in critical but stable condition.
Saipov had spent several weeks preparing the attack and precisely followed the online instructions of the Islamic State terror organization, authorities in New York said on Wednesday in a press conference.