NEW YORK — In a party that’s shifted leftward on abortion rights, Democratic presidential hopefuls are offering different approaches to a central challenge: How to talk to voters without a clear home in the polarizing debate over the government’s role in the decision to end a pregnancy.

While Bernie Sanders said this month that “being pro-choice is an absolutely essential part of being a Democrat,” his presidential primary opponent Amy Klobuchar took a more open stance last week in saying that pro-life Democrats “are part of our party.” Klobuchar’s perfect voting score from major abortion-rights groups makes her an unlikely ally, but some abortion opponents nonetheless lauded the Minnesota senator for extending a hand to those on the other side of an issue that’s especially important for Catholics and other devout voters.

The praise for Klobuchar suggests that Democrats who have heeded rising worry within their base about GOP-backed abortion limits by pitching significant new abortion-rights policies may risk alienating religious voters who are otherwise open to supporting their party over President Donald Trump. Voters in that group looking for an appeal to “common ground” on abortion, as former President Barack Obama put it during his 2008 campaign, have heard few of those statements during the current Democratic primary.

“Plenty of pro-life Catholics are looking for an alternative to voting for President Trump,” said Kim Daniels, associate director of Georgetown University’s Initiative on Catholic Social Thought and Public Life. “We wish the Democratic Party would offer us an alternative instead of doubling down on support for abortion throughout pregnancy, taxpayer funding and the like.”

Klobuchar has underscored her abortion-rights support, and she’s signed onto legislation that would limit states’ efforts to constrain abortion access, such as the multiple state-level anti-abortion laws that passed last year. But Daniels described Klobuchar’s rhetorical openness to working with abortion opponents as “an important step,” and she’s not alone.

Chris Crawford, a pro-life activist who tweeted about Klobuchar’s welcoming response to him during a recent event in New Hampshire, said that “I don’t like” the senator’s abortion record or positions, “but I do like the work she’s doing on adoptions.”

“And if she’s serious about putting together an agenda that can provide for mothers … that would make a big difference for me and other voters I know,” added the Catholic Crawford, who said he voted for Hillary Clinton in 2016 but has not yet decided who he’s supporting in 2020.

Religion is not the only factor motivating potential Democratic voters who favor some degree of abortion limits — Democrats for Life executive director Kristen Day pointed out in an interview that atheists are part of her coalition. But abortion restriction is still a priority for a sizable number of Catholics, even as Pope Francis orients the church toward a more expansive definition of the term “pro-life,” pressing President Donald Trump on some of his immigration policies.

An AP-NORC poll taken in December found that 45 percent of Catholics backed significant restrictions that would make abortion illegal except in cases of rape, incest, or threats to a mother’s life. Among Democrats and Democratic-leaning adults, 17 percent said that abortion should be illegal in most or all cases, a number that rises to 25 percent among self-identified conservative or moderate Democrats, according to a Pew Research Center survey last year.

The abortion debate is set to return to the political forefront next month, when the Supreme Court hears arguments in a high-profile challenge to a Louisiana state law, authored by a pro-life Democratic lawmaker, which requires doctors performing abortions to have admitting privileges at a nearby hospital. A final decision is anticipated by June.

Charles Camosy, an associate professor of theology at Fordham University who recently left the board of Democrats for Life in frustration over what he saw as the party’s absolutist approach to abortion, asserted that “something is missing” when the same blanket “pro-choice” terminology can be used to apply to both Klobuchar and Sanders.

A Democratic candidate willing to focus on common ground could have “a golden opportunity to meet pro-lifers, or at least religious people who are mildly pro-choice,” Camosy said.

However, Klobuchar’s comments left some abortion-rights and pro-life activists cold. The pro-life group Susan B. Anthony List tweeted that the Minnesota senator is “still extreme & out-of-touch,” pointing to her record of abortion-rights votes, and Ilyse Hogue, president of the abortion-rights group NARAL Pro-Choice America, warned Klobuchar against “giving credence to right-wing red herrings.”

Hogue said in an interview that under the umbrella of abortion-rights advocacy, she sees room for shared values of “compassion and freedom” as well as different feelings about the decision to terminate a pregnancy.

“Where we are in common service together,” Hogue said, is that “none of us ever want anyone to feel like they have to terminate a pregnancy because they will not get the support they need to parent.”

Hogue also underscored the sharp contrast between Democrats and the GOP, where Trump has embraced anti-abortion policies and burnished his standing with religious conservatives as a result. That distance between the parties has grown in recent years, with fewer pro-life Democrats serving in Congress and two straight Democratic platforms adopting stronger language on abortion rights.

Indeed, Sanders described abortion-rights support as “essential” this month but took flak from some abortion-rights advocates in 2017 for backing a pro-life Catholic candidate for mayor of Omaha, Nebraska. Pro-life Democrats are not wholly extinct, with the Catholic Louisiana Gov. John Bel Edwards winning reelection last year, but two such members of Congress are facing primary challenges from the left.

In this year’s Democratic presidential primary, Klobuchar’s inclusive language marked a rare instance of daylight between candidates in an abortion debate that’s already put pressure on her rivals.

When pressed on abortion by executive director Day of Democrats for Life during a Fox News town hall last month, former South Bend, Indiana, Mayor Pete Buttigieg said that “I support the position of my party that this kind of medical care needs to be available to everyone.”

Joe Biden, a Catholic who last year reversed his stance to back unrestricted federal funding for abortions, was denied communion by one South Carolina priest last fall in response to the former vice president’s support for abortion rights.

Massachusetts Sen. Elizabeth Warren, for her part, said at November’s debate that safeguarding abortion rights is “fundamentally what we do and what we stand for as a Democratic Party” and demurred when pressed about whether an abortion opponent like Bel Edwards would be welcome.

Like those Democrats, Klobuchar supports codifying the abortion-rights protections of the Supreme Court’s Roe v. Wade decision into law. And for some pro-life voters, inclusive language like Klobuchar’s may not be enough to overcome that substantive Democratic alignment.

Klobuchar’s handling of the issue is “going to make her look much more moderate” and could break through with potentially persuadable Catholic voters, said Robert George, a Princeton University professor and past GOP appointee to the U.S. Commission on International Religious Freedom.

But that advantage can be “undercut” by Klobuchar’s abortion-rights votes, George added, which Trump’s campaign would seek to do.

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