The president was considering the move in the wake of the cycle path rampage which left eight people dead and at least 12 injured.
Islamic State fanatic Saipov, a 29-yearold migrant from Uzbekistan, was said to be “gleeful” as he recovered in hospital from police bullet wounds in his stomach.
He had pledged allegiance to the terror network in a note found inside the rented van used to speed down a riverside cycle path.
Asked whether the controversial camp in Cuba was an option for Saipov, Mr Trump said: “I would certainly consider that. Send him to Gitmo.
“We also have to come up with punishment that’s far quicker and far greater than the punishment these animals are getting right now.”
Earlier Republican senator and former US Air Force colonel Lindsey Graham said that Saipov, a married father of three, should be treated as an enemy combatant.
The president also used Tuesday’s attack to call for tighter migration controls after it emerged that Saipov was allowed into the US through a visa “lottery” scheme.
Saipov emigrated from Uzbekistan seven years ago, worked as an Uber cab driver and was unknown to police except for traffic infringements.
But New York Deputy Police Commissioner John Miller revealed that he is suspected of carefully planning New York’s worst terrorist incident since 9/11.
Mr Miller said: “It appears that Mr Saipov had been planning this for a number of weeks.
“He did this in the name of IS and along with other items recovered at the scene were some notes that indicate that.
“He appears to have followed almost exactly the instructions that IS has put out on its social media channels to its followers.”
New York Governor Andrew Cuomo said Saipov was “radicalised domestically” at a mosque with a reputation for hardline Islam.
Saipov is said to have fled the van following the attack on the west side of lower Manhattan, just a few blocks from the scene of the 2001 Twin Towers outrage.
Witnesses said the driver chanted “Alahu Akhbar” – God is greatest in Arabic – before emerging, with two weapons which proved to be a pellet gun and a paintball gun.
After he was shot in the stomach by 28-year-old NYPD officer Bryan Nash, police found an IS flag and a note saying “IS lives forever” in the van.
The suspect underwent surgery at the city’s Bellevue Hospital, and was reported to be “proud” and “boasting” of the massacre when interviewed by the authorities.
Saipov’s victims included a mother of two from Belgium and five Argentinian college reunion friends. Anne-Laure Decadt, 31, was hailed a “beautiful mom” to sons aged three and two months by her husband Alexander Naessens.
Tributes also poured in for the group of South American architects and businessmen Ariel Erlij, Hernan Diego Mendoza, Diego Enrique Angelini, Alejandro Damian Pagnucco and Hernan Ferruchi.
They were enjoying a cycling trip around Manhattan to mark the 30th anniversary of their graduation from a technical college.
The six were named among eight who died. Mr Naessens said his wife, who was on holiday with her mother and two sisters, was cycling along the route when Saipov ploughed into her from behind.
He said: “She was a wonderful wife and the most beautiful mum of our two-month and three-year-old sons.
“The loss is unbearable and cannot be overcome.”