President-elect Joe Biden has promised to take dramatic action to address the most pressing issues of American life. But the degree that Congress will be part of that effort is likely to depend on a small group of Republican senators and the improbable return of a bygone era of bipartisan cooperation.

Yet if anyone can manage it, senators and operatives from both parties say, it’s Biden — who spent 36 years in the Senate and will enter the White House as the most accomplished legislator to hold the presidency since at least Lyndon B. Johnson.

Those skills will face a daunting test after Biden’s presidential victory failed to produce major Democratic gains in the Senate, leaving control of the chamber dependent on the outcome of a pair of Jan. 5 runoff elections in Georgia. Many on Capitol Hill are already gaming out scenarios for at least two years of divided government, with Biden forced to wrangle with his former colleague, Senate Majority Leader Mitch McConnell (R-Ky.).

At stake is not only Biden’s policy agenda but the process of filling his Cabinet and other agency posts, as well as the increasingly partisan business of judicial nominations. How Biden navigates the political currents of the Senate could determine such essential matters as the federal government’s fiscal outlook or filling a possible Supreme Court vacancy.

The scenarios tend to have Biden wooing a small number of Republicans with a reputation for bipartisanship — centered on Sens. Susan Collins of Maine and Lisa Murkowski of Alaska — as he seeks to build a governing coalition in a closely divided Congress. Both Republicans offered encouraging words in the days following Biden’s victory.

“Joe Biden knows exactly how to deal with this body. He’s worked within it. He has done some strong and important bipartisan deals with no less than Mitch McConnell,” Murkowski said in an interview Thursday. “If you truly have a view that what we’re going to try to do is be constructive rather than just throw grenades at one another, you can build anything.”

Biden phoned Collins — who handily won her own reelection race despite tens of millions of dollars of Democratic spending — shortly after Election Day, and she hailed her “very good relationship” with the president-elect in comments to reporters this week.

“As I would of any president, I want him to be successful,” she said Tuesday.

Weighing against the happy talk is recent history in the Senate, where groups of moderate senators have joined over the past decade to try to solve intractable issues such as illegal immigration, gun violence and a spiraling national debt with scant success.

A group that met in Collins’s office amid a government shutdown in early 2018, for instance, tried to forge a path to an immigration deal that never materialized. Bipartisan talks on modest gun control measures led by Sen. Joe Manchin III (D-W.Va.) and Sen. Patrick J. Toomey (R-Pa.) have gone nowhere. Compromises struck by bipartisan “gangs” to preserve the rights of Senate minorities have been obliterated.

To many Democrats — who vividly remember McConnell’s efforts to block President Barack Obama’s legislative priorities as both minority and majority leader — there is simply no hope of progress as long as he sets the Senate agenda.

“We saw that horror film play out in the Obama years, and we don’t want to see it again,” Sen. Jeff Merkley (D-Ore.) said. “Unless there’s a group secretly meeting within the Republican caucus that says, ‘We really want to see the Senate work as a legislative body again,’ then my hopes are dim if Leader McConnell’s in charge.”

Ronald A. Klain, Biden’s choice for White House chief of staff, said in an MSNBC interview Thursday that Biden had yet to speak to McConnell post-election but noted the two men have known each other for decades, with a track record of cutting deals.

Klain expressed measured optimism that they “will have a working relationship when the time comes,” even though McConnell has yet to publicly acknowledge Biden’s win a week after the Associated Press and multiple news organizations called the race for him.

“They obviously need no introduction to one another,” he said, adding that Biden would “use all the tools that he has, his prodigious tools of persuasion, his ability to reach out to people on all aspects of our party and in both parties, to rally the voters and to do everything he can to move this country forward.”

The early part of that effort has included conversations with Republican senators — few of whom have publicly recognized Biden’s victory as Trump has refused to concede. Yet the power of personal relationships has drastically receded in the 11 years since Biden left the Senate, and his former colleagues warn that he will be battling fierce institutional obstacles.

Former senator Kent Conrad (D-N.D.), who served 22 years with Biden, said the days of bottom-up lawmaking driven by committees and bipartisan coalitions of senators are “largely gone.”

“The ability of the majority leader to set the agenda is incredibly important. It gives him extraordinary leverage. And that, more than anything else, probably will dictate what the Biden legislative agenda is going to be,” Conrad said.

Democrats cling to the possibility that they might still be able to sideline McConnell through the Georgia runoffs. They would need to win both races to force a 50-50 split, creating a governing majority with Vice President-elect Kamala D. Harris’s tie-breaking vote. In that case, power would shift to a cadre of Democratic moderates — such as Manchin and Sen. Kyrsten Sinema of Arizona — who would effectively define the limits of Biden’s policy ambitions.

But many Democrats are pessimistic because more Republican votes were cast in both Georgia races despite Biden’s presidential win, and the GOP has increased its vote share in seven of the last eight statewide runoffs in Georgia dating back three decades.

Should Biden become the first president since George H.W. Bush to enter office without the Senate under his party’s control, the mere task of confirming a Cabinet threatens to become a partisan minefield, handing Republicans effective veto power over his administration’s staffing.

A clutch of GOP senators already signaled that they are likely to vote to confirm Biden’s nominees, but under certain conditions. Collins said she would have reservations if a nominee is “not qualified or is way outside the ideological mainstream or is unethical,” while Sen. Mitt Romney (R-Utah) — who has shown a willingness to cross party lines on some issues — said he would confirm Biden’s nominees “barring some extreme individual or someone not qualified for the job.”

While McConnell might be unlikely to orchestrate a blockade of Biden’s Cabinet nominees, he is also unlikely to countenance any legislative deals the new president may seek to strike with two or three members of his conference. With a 60-vote threshold for major legislation all but certain to remain in place, any legislative negotiation will have to include a larger group of Republicans.

In one vivid illustration of his aversion to legislation that divides his ranks, McConnell refused for months to schedule a vote on a bipartisan criminal justice reform bill until Trump publicly pushed him. It ultimately passed 87 to 12.

Josh Huder, a congressional scholar and senior fellow with Georgetown University’s Government Affairs Institute, said it “seems improbable at best” that governing from the center would succeed in a hyperpartisan era.

“In the last decade and a half of divided control of Congress, we’ve seen historic gridlock — the kind of stalemate where you can’t pass a budget or appropriations bills, much less touch on some of the most high-profile, high-stakes political battles that we have,” he said.

Even those senators most likely to cooperate with Democrats are warning their cooperation will only go so far. Murkowski, who is up for reelection in 2022 and whose state depends on oil and gas jobs, warned, “If there are issues that work against my state’s economy, we’re going to fight them tooth and nail” — a dismal indicator for climate-change legislation.

Romney said Biden would be “making a mistake to think that the people of America chose a left-oriented policy agenda” given the mixed election results. “The country remains a center-right nation, and I will continue to fight for conservative principles that I think comport with not only my philosophy but that of the nation as a whole,” he said.

Sen. Christopher A. Coons (D-Del.), a close Biden ally, said he did not expect the president-elect to be focused solely on outreach to Collins, Murkowski and the handful of other moderates, but to a larger group of senators across party lines.

“He recognizes, I believe, how central that is to making progress,” he said, adding: “The last four years would give no one objectively hope that Mitch McConnell will be a constructive, bipartisan legislator, but he’s been a senator a very long time, and he has had periods when he was, and I have to remain hopeful that he may yet again.”

Rachael Bade, Seung Min Kim and Erica Werner contributed to this report.