WASHINGTON — Democrats rang alarm bells on Sunday about the likelihood that Republicans would try to restrict abortion nationwide, two days after an interview was published in which Senator Mitch McConnell of Kentucky, the minority leader, said a ban was “possible” if his party gained control in Washington.

On the Sunday talk shows and in other public statements, Democratic senators said Republicans would not stop at letting the states decide the issue, but would most likely push for federal restrictions. That made it paramount, they said, that the Democratic Party maintain control of the Senate as it tries to codify abortion rights into federal law.

“We need to make sure that every single voter understands that the Republican Party and Mitch McConnell does not believe that their daughters, that their mothers, that their sisters have rights to make fundamental life and death decisions,” Senator Kirsten Gillibrand, Democrat of New York, said on CNN’s “State of the Union.” “We are half-citizens under this ruling. And if this is put into law, it changes the foundation of America.”

After a leaked draft decision indicated that the Supreme Court was poised to overturn Roe v. Wade, the landmark 1973 case that established a constitutional right to abortion, Mr. McConnell said in an interview with USA Today that a national abortion ban was “possible” if that draft document became an official opinion of the court.

“If the leaked opinion became the final opinion, legislative bodies — not only at the state level but at the federal level — certainly could legislate in that area,” Mr. McConnell said when asked if a national abortion ban was “worthy of debate.”

“So yeah, it’s possible,” he added.

Mr. McConnell argued that the discussion about a federal ban was premature, but that it was clear that the Republican Party has long been opposed to abortion. Discussions are already underway among some Republican senators about pushing to ban abortion after a certain number of weeks, ranging from six to 20, depending on the proposal.

“If and when the court makes a final decision, I expect everybody will be more definitive,” Mr. McConnell said. “But I don’t think it’s much secret where Senate Republicans stand on that issue.”

A document circulated by the National Republican Senatorial Committee and obtained by Axios urged candidates to be low key about abortion, casting themselves as “compassionate consensus builders” with a post-Roe America looming as early as next month.

“States should have the flexibility to implement reasonable restrictions,” the document states.

Senator Lindsey Graham, Republican of South Carolina, said on “Fox News Sunday” that he would push a bill to outlaw abortion nationwide after 20 weeks of pregnancy.

“We’re one of the seven nations in the world that allow abortion on demand at 20 weeks, the fifth month in pregnancy,” Mr. Graham said. “Congress will continue to debate this issue.”

But Senator Amy Klobuchar, Democrat of Minnesota, said on ABC’s “This Week” that no woman wants a Republican senator — like Ted Cruz of Texas — deciding whether she can have an abortion.

“Who should make this decision? Should it be a woman and her doctor or a politician? Should it be Ted Cruz making this decision or a woman and her family?”

Senator Maggie Hassan, a New Hampshire Democrat who is up for re-election, blasted out a news release of Mr. McConnell’s comments on Saturday evening.

“Mitch McConnell is already making it clear that if Republicans retake the Senate majority, they will quickly move to criminalize abortion nationwide and roll back reproductive freedom for women all across this country, including in New Hampshire,” Ms. Hassan said.

Senator Chuck Schumer, Democrat of New York and the majority leader, has set up a Wednesday vote on a bill to codify abortion rights into federal law. The legislation is all but certain to be blocked by Republicans, falling short of the 60 votes needed to advance past the Senate’s legislative filibuster.

It also appears to lack even the simple majority it would need to pass the 50-50 Senate, given that Senator Joe Manchin III, the centrist Democrat from West Virginia who opposes abortion rights, voted against bringing up a nearly identical measure in February and showed no signs that he had shifted his position.