As moderate Democrats who might be willing to work with Republicans on some issues, Sens. Heidi Heitkamp of North Dakota and Joe Manchin III of West Virginia might be some of the Senate’s most influential members in the new Congress. (Melina Mara/The Washington Post)

Some of the most influential senators in the new Congress are neither in the majority nor among the longest-serving. They don’t show up on the Sunday-morning talk shows, and they aren’t talking about running for president in 2016.

Instead, they’re a pack of Democrats from mostly smaller, rural states who are inclined to work with Republicans on legislation President Obama doesn’t support. They may even be willing to help the GOP override his vetoes.

Some of them support building the Keystone XL oil pipeline and are expected to be active as the Senate begins to debate the issue this week. Others want Congress to pass tougher sanctions against Iran, and all are open to making changes to Obama’s health-care law. All three issues have drawn veto threats from the White House in recent days.

One of the biggest unanswered questions about the week-old Congress is whether the new Republican majority will be able to overcome Capitol Hill’s culture of stifling partisanship and cultivate enough Democratic support to challenge Obama.

These moderate Democrats say they will cooperate if Republicans don’t use the Senate floor to score political points — as Democrats have done over the past several years. They have big expectations for Senate Majority Leader Mitch McConnell (R-Ky.), who will need to keep his larger conference unified while sustaining his promise to allow a more open and nonpartisan debate process.

Sen. Joe Donnelly (D-Ind.), who is firmly planted in the middle of both parties, said he’ll support bills “that both Republicans and Democrats who sit down and talk to each other think will make things better for us. I think you’ll see that extreme legislation — whether right or left — is going to go nowhere.”

The moderates’ influence could get an early test this week as the Senate begins debating the Keystone pipeline again. Despite an Obama veto threat, the House voted Friday to authorize the long-delayed project, with the support of 28 moderate Democrats. McConnell has promised a Keystone debate of several days, with time set aside to debate proposed amendments from both parties. Passage of those amendments likely will rest with moderate Democrats.

“There isn’t any particular issue that cries out for bipartisanship — all of the issues do,” said Sen. Heidi Heitkamp (D-N.D.), who strongly supports the Keystone project. “The challenges that we will have is making sure that we don’t once again spiral into the thinking that it’s okay to delay decisions, that it’s okay not to do the work of the Senate.”

Sen. Bob Corker (R-Tenn.) spent his first eight years in the Senate confined to the minority and sounded envious of the new influence of some Democrats.

“When you’re in the majority and you’re rank-and-file, you’re taken for granted,” he said in a recent interview. “When you’re in the minority, if you’re somewhat interested in advancing our nation, you actually end up being paid far more attention than if you’re in the majority.”

Working with moderate Democrats will be critical for Republicans as they retake control of Congress, Corker said. “We’re going to be getting a report card very quickly from the American people regarding how we are able to legislate.”

Senate Minority Leader Harry M. Reid (D-Nev.) and his leadership team plan to give the moderates time and space to work with Republicans on some issues, especially since many of them ran on a promise to do so.

But that doesn’t mean Reid won’t play hardball. He warned last week that “any attempt to erode protections for working American families . . . will be met with a swift and unified Democratic opposition.”

Conversations with more than a dozen senators and senior aides in both parties revealed that several moderate Democrats are likely targets as Republicans begin work on their early legislative priorities.

Three names came up in every conversation: Donnelly, Heitkamp and Sen. Joe Manchin III (D-W.Va.), a former governor who got his first taste of high-stakes dealmaking during the 2013 gun-control debate. The next mentioned most are a trio of former governors, Sens. Timothy M. Kaine (D-Va.), Angus King (I-Maine) and Mark R. Warner (D-Va.), who are from states that expect their lawmakers to negotiate with the other side.

From there, other Democrats are “gettable” — a term used by many senators and aides — on a variety of issues. Sens. Michael F. Bennet (Colo.), Robert P. Casey Jr. (Pa.), Christopher A. Coons (Del.), Martin Heinrich (N.M.), Amy Klobuchar (Minn.), Claire McCaskill (Mo.), Jeanne Shaheen (N.H.) and Jon Tester (Mont.) were all mentioned as amenable to working on energy, economic, national security and tax issues.

So is Sen. Thomas R. Carper (D-Del.), who earned credit for working with House and Senate Republicans last year on tedious government management bills. Sen. Robert Menendez (D-N.J.) is also expected to stand with GOP hard-liners against Obama’s overtures to Cuba.

“There will be a group of about 15 Democrats who are willing to work with [Republicans] on any given issue,” Kaine predicted in an interview. He is eager to work with the GOP on authorizing military action against the Islamic State and to revamp presidential war powers.

Fifteen is an encouraging number to Republicans, who now have 54 seats. If every Republican agrees on a piece of legislation, they will need to find at least six Democrats to help a bill clear the chamber’s arcane procedural hurdles and pass. If a few Republicans peel away — which is likely given the inflexibly conservative views of some GOP senators — McConnell might need to rely on even more Democrats. Sixty-seven senators will be needed if Republicans want to override an Obama veto.

The White House has said that Obama also opposes a bill co-sponsored by Donnelly and Sen. Susan Collins (R-Maine) that would change the Affordable Care Act’s definition of “full-time employment” back to 40 hours per week.

Donnelly dismissed Obama’s veto threat: “In Indiana and every other state, 40 hours is full time. It’s just the way things are in America.”

He said he wants to fix the health-care law, not repeal it: “How do you explain to somebody with diabetes or arthritis or a heart condition who’s received health care for the first time in their life maybe that we’re going to take it away from them? I don’t want anything to do with that.”

Heitkamp predicted that the Keystone plan will pass in the Senate but that it will take longer to find senators willing to override Obama’s veto.

“This first go-round we’re not going to have 67,” she said. “We’ll probably have something in the neighborhood of 63. So we’ll need to find four more votes. The question will be, would members be willing to reconsider a ‘no’ vote based on other priorities? That’s the art of legislating.”

Manchin, who was elected to the Senate in 2010, said he’s eager to cut new deals in the next two years, because “the first four years wasn’t that exciting.” He confirmed that he’s thought about running for governor again in 2016, but if the dealmaking intensifies, “it makes it much more attractive for me to stay here and keep fighting the fight.”

Heitkamp did not deny that she is also thinking about running for governor in 2016, but she added, “When I feel like we’re making timely, effective decisions, that’s when there’s going to be job satisfaction.”

King is staunchly opposed to the Keystone project. “It’s not the role of Congress to be issuing building permits,” he said. But he wants to work with Republicans on other energy issues and proposals to revamp the student loan system and cybersecurity policy.

“It’s not like this is an auction process where I’m saying I’m for sale,” he said. “But I think and hope that I’ve already established a reputation as someone who listens and tries to go for what’s sensible.”

As the chamber convened last week, virtually every senator seemed to have a bipartisan idea. Sens. Dianne Feinstein (D-Calif.) and Lindsey O. Graham (R-S.C.) introduced a bill to crack down on animal poachers. Shaheen and Sen. Rob Portman (R-Ohio) unveiled a new version of their long-sought energy-efficiency legislation. Klobuchar and Sen. John McCain (R-Ariz.) announced plans for legislation that would permit Americans to import prescription drugs from Canada.

Those are exactly the kinds of ideas and partnerships that bring the moderates a renewed sense of optimism.

“There’s going to have to be some venting on both sides, but if that can be done in a way that doesn’t lead to everybody going back to their foxholes, that would be really good for the country and for Democrats as well,” Warner said.

“It’s restart season,” he added. “Trying to figure out ways to get to ‘yes’ is kind of an exciting opportunity.”