Trump said Saipov came to the US through the visa programme.
Trump also said he would 'certainly consider' sending Saipov to America's military detention centre in Guantanamo Bay, Cuba.
"I would certainly consider that," Trump told reporters.
Trump had earlier lashed out at the 'Diversity Visa Lottery Program' blaming the Democrats and pressed for a 'merit-based' program for immigrants to the United States.
"The terrorist came into our country through what is called the 'Diversity Visa Lottery Program', a Chuck Schumer beauty. I want merit-based," Trump wrote in a post on Twitter, referring to the Senate Democratic leader.
Meanwhile, New York police said Sayfullo Saipov was planning the attack for weeks.
"Based on the investigation overnight, it appears that Mr Saipov had been planning this for a number of weeks," John Miller, deputy commissioner of the New York Police Department said.
The police said Arabic notes referencing Islamic State were found at the New York attack site.
Police also added that Saipov was interviewed at the hospital but did not give details.
Sayfullo Saipov had never been the subject of a federal or local police probe, John Miller, deputy commissioner of the New York Police Department told reporters.
Mayor Bill de Blasio told a news conference that the New York Marathon will be held as planned. He said security will be heavily stepped up for the race involving more than 50,000 runners.