NEW YORK - As flags fly at half-staff at the White House and Supreme Court, the debate heats up over when a replacement for Supreme Court Justice Antonin Scalia should be nominated.

Senate Majority Leader Mitch McConnell (R-Kentucky) says, "The American people should have a voice in the selection of their next Supreme Court justice." 

But the top Democrat on the Senate Judiciary Committee, Vermont Sen. Patrick Leahy, told CNN's "State of the Union" that McConnell is "certainly ignoring the Constitution." 

"The Senate has a responsibility to fill vacancies as soon as possible," says Minority Leader Sen. Harry Reid (D-Nevada).

Sen. Chuck Schumer (D-New York) has also weighed in. "The American people don't like this obstructionism," said Schumer. "When you go right off the bat and say, 'I don't care who he nominates, I am going to oppose him,' that's not going to fly."

President Obama acknowledged Scalia's death and promised to nominate a replacement soon. "I plan to fulfill my constitutional responsibilities to nominate his successor in due time," said Obama. "There will be plenty of time for me to do so and for the Senate to fulfill its responsibility to give that person a fair hearing and a timely vote."

Scalia, whose body was back in Virginia Monday, died over the weekend in Texas at age 79 of natural causes, and candidates at a Republican presidential debate also said Obama shouldn't nominate the next justice. Donald Trump suggested that Republicans should "delay, delay, delay" on a nomination.

Obama has vowed to wait until Congress is back in session before choosing a nominee. If Republicans are able to delay the process until Obama leaves office, it would be a record. On average, it took around 70 days to confirm current members of the court.

Associated Press reports contributed to this story.