U.S. President Donald Trump on Wednesday urged tougher immigration measures after the previous day's deadly truck attack in New York City, referring to the suspect as "this animal."

Trump said he will ask Congress to eliminate the country's diversity visa program.

"Sounds nice, but it's not nice," said Trump, making a case for merit-based immigration and the elimination of chain migration, where arrivals enter through family connections in the U.S.

Trump made his comments to reporters in a freewheeling address prior to a meeting with his cabinet at the White House.

Trump earlier in the day on Twitter said that the driver in Tuesday's attack "came into our country through what is called the 'Diversity Visa Lottery Program,' a Chuck Schumer beauty" — a reference to the Senate's top Democrat.

Schumer fired back from the Senate floor, accusing Trump of "politicizing" the tragedy.

White House spokeswoman Sarah Huckabee Sanders said Wednesday that Trump's calls for increasing extreme vetting of immigrants are "something that, frankly, the president has been talking about for a long time."

Sanders told reporters the president "has not blamed" the New York Democrat and "doesn't feel that the senator is responsible for the attack."

Sen. Jeff Flake, a Republican from Arizona, said on Twitter that Trump was unfairly blaming Schumer for the diversity visa program.

Flake, one of Trump's chief Republican foes in Congress, said Schumer was among a group of eight Republican and Democratic senators who proposed eliminating the program three years ago as part of a broader bipartisan bill to overhaul U.S. immigration laws.

Flake, who served on that "Gang of Eight" with Schumer, said: "I know. I was there."

The immigration bill ultimately failed in the Republican-led House after passing the Senate in June 2013, 68-32, with 14 Republicans joining Democrats.

Republican Sen. Jeff Flake, who promised to speak freely after announcing he won't run in next year's midterms, quickly corrected the president on his interpretation of a past legislative effort regarding visas. (Manuel Balce Ceneta/Associated Press)

Flake recently announced he's not running for re-election but says he won't be silent about Trump's politics and behaviour.

Officials said the suspect in the attack is an immigrant from Uzbekistan who came to the United States legally in March 2010.

"According to reports to my office, [Sayfullo] Saipov is a lawful permanent resident who immigrated to the United States from Uzbekistan on a diversity visa over five years ago," Senate judiciary committee chairman Chuck Grassley wrote in a letter to the secretary of state and the acting secretary of Homeland Security.

Uzbekistan not on travel ban list

The visa program provides up to 50,000 visas annually by lottery. Applicants must have a high school diploma or meet work experience requirements. It was created as part of a bipartisan immigration bill introduced by the late Ted Kennedy, a Democratic senator who represented Massachusetts, and signed into law by then-president George H.W. Bush in 1990.

House Judiciary Committee chairman Bob Goodlatte, a Republican, was also quick to call for the visa program to be terminated.

'The president's tweets were not helpful,' says Gov. Andrew Cuomo 3:28

But Peter King, a Republican who's been an ardent supporter of broad anti-terror powers since 9/11, told CNN in an interview that the lottery system on the whole has worked to bring "good people" into the country.

Trump was only considering one party in his remarks before meeting with his cabinet, however.

"Honestly, they don't want to do what's right for our country," Trump said of the Democrats.

During the 2016 presidential campaign, Trump proposed a total ban on Muslim immigration to the U.S., before embracing "extreme vetting." Trump's efforts to block immigrants from several Muslim-majority countries, as well as his plans regarding the U.S.-born children of illegal immigrants, have been tied up in litigation in federal courts.

Uzbekistan has never been on the list of countries in any of Trump's proposed travel bans.

Schumer hits back

Speaking on the Senate floor Wednesday, Schumer said he has "always believed that immigration is good for America." He also criticized Trump for "politicizing" the deadly attack.

"President Trump, where is your leadership?" Schumer said. "The contrast between President [George. W.] Bush's actions after 9/11 and President Trump's actions this morning could not be starker."

Senate minority leader Chuck Schumer took aim at Trump for 'politicizing' the deadly bike path attack in New York City. (Aaron P. Bernstein/Reuters)

He said Trump had proposed cutting anti-terrorism funding in his most recent budget.

"I'm calling on President Trump to rescind his proposed cuts to this vital anti-terrorism funding immediately," Schumer said.

Critics have taken note of Trump's immediate tweets and statements in the wake of Monday's attack, in contrast to a relatively muted response on social media, beyond extending condolences, when a white, U.S.-born shooter killed 59 people and wounded hundreds in Las Vegas in October.

In addition, Sanders said the day after the Las Vegas shooting that "there's a time and place for a political debate, but now is the time to unite the country."

New York Gov. Andrew Cuomo, a Democrat, didn't hold back when asked about Trump's running social media commentary after the attack. He called into question the accuracy of Trump's comments, saying they were "not helpful" and "had no relevance to the facts of this situation."