In recent years, Sens. Lamar Alexander (R-Tenn.), left, and Charles Schumer (D-N.Y.) have arguably developed the most important friendship in the upper chamber. ((AP/AP))

In the next few days, the Senate will begin debate on the reauthorization of a child-care development block-grant program — a small, relatively noncontroversial program that is likely to pass.

In this case, it’s not the “what” of the effort that’s most important but the “how.” The debate is expected to take a different course, with both sides getting equal time to make arguments and offer amendments — a practice that has pretty much all but disappeared in the increasingly partisan Senate.

Majority Leader Harry M. Reid (D-Nev.) is expected to cede control of the floor to the bill’s lead sponsors, Sens. Barbara A. Mikulski (D-Md.) and Richard Burr (R-N.C.), who will lead the debate and allow Democrats and Republicans to introduce amendments.

More important — if it works — Reid is expected to permit more debate on other bipartisan measures on subjects ranging from sentencing overhauls to manufacturing and energy efficiency. In some small ways, the Senate may begin to look more like the great deliberative body it’s supposed to be.

Two people who get much of the credit for this development are Charles E. Schumer (D-N.Y.) and Lamar Alexander (R-Tenn.), who arguably have developed the most important friendship in the Senate in recent years.

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During several weeks in January and February, aides said, Schumer and Alexander quietly orchestrated what both described as a “modest experiment” based on a simple premise: Senators should be able to debate, amend and pass legislation supported by members of both parties.

“I’ve only been here 14 years, and Alexander’s been there about 11,” Schumer said. “But we were there, both of us were there and remember when the Senate used to legislate, and thoroughly enjoyed it and wish it would return.”

In a separate interview, Alexander said that they are “trying to start a week focused on what you can do, not what you can’t do.”

“This requires restraint by all senators,” he said. “But occasionally we ought to try.”

When Reid and Minority Leader Mitch McConnell (R-Ky.) openly feuded on the Senate floor over arcane procedures, it often fell to Schumer and Alexander to huddle in a corner, come up with a proposed solution and quietly sell the idea to party colleagues.

The pair succeeded, with the help of other senior senators, in averting several meltdowns. But they couldn’t avert the decision by Democrats in November to change the rules to make it easier for them to confirm President Obama’s nominees by ending the 60-vote threshold for filibusters on those nominations, and depriving Republicans of the leverage to block those nominations.

In the months since, Republican senators have blocked consideration of most legislation and slowed the process of confirming dozens of Obama nominees. The GOP bitterness has been palpable.

Alexander, a defender of Senate tradition, was especially aggrieved by what he described as an “abhorrent” change in procedure. And Schumer, eager to help Democrats score legislative victories, wanted to find a way to heal partisan wounds and get bills moving again.

The unlikely partnership between Schumer and Alexander developed over several years. Together they led the Senate Rules Committee, giving them a say in the planning for the presidential inauguration. Both eager to tout their home states during last year’s ceremony, Alexander asked a Tennessee college choir to perform, while Schumer invited the Brooklyn Tabernacle Choir and served New York wines and spring water at the inaugural luncheon.

Alexander shocked his colleagues in 2011 by stepping down as head of the Senate GOP messaging office in hopes of finding ways to help restore bipartisan cooperation. In the years since, he said, “I feel I provide more leadership outside the elected leadership.”

Schumer’s outreach to Republicans is part of an unlikely transformation fraught with political risk as he attempts to position himself as the likely heir to Reid. After winning a tough 2010 reelection, Reid is indisputably the Democratic leader for the next two to four years. But a long list of deaths and retirements has altered the Senate’s composition, leaving it short of elder statesmen with the gravitas to be honest brokers.

Ultimately, Schumer and Alexander said, they hope their experiment will help the Senate return to regular order, although Alexander admits that he’s skeptical. “We’ve had so many things that haven’t worked that I’m going to keep my expectations low and be pleasantly surprised,” he said.

Mikulski, who wrote the appropriations bill that passed the Senate with 72 votes in January, is especially excited at the prospect of bipartisan progress. With 45 senators serving their first term, she said, the exercise will be critical to ensuring the future of the Senate. “Many members who’ve come here since 2006 and 2008 have no idea when we say what regular order is,” she said. “To them, regular order was chaos, confrontation and cloture votes.”

Conscious that the scars of old political fights don’t heal overnight, Schumer is trying to be more socially engaged with Republicans. He has become a regular in warmer months on the Black Tie, a boat owned by Sen. Joe Manchin III (D-W.Va.). Manchin uses the vessel as an off­-the-record spot for bipartisan gatherings. Some junior Republicans initially refused Manchin’s invitation when they heard Schumer was going to be there.

Last year, Manchin recalled, one Republican had said, “I don’t want to go if he’s going to go.” He had to coax, but most of the Republicans have found Schumer to be more likable than they imagined, Manchin said. “People that make the effort find there’s someone to work with.”

Schumer’s outreach to Republicans has been greeted with some skepticism. In an interview in the summer, shortly after the Senate approved a bipartisan immigration reform plan and an earlier parliamentary showdown was averted, Alexander said Schumer had “built up a lot of ill will” with Republicans after years as a leading combatant in the nonstop partisan wars on Capitol Hill.

“He’s worked hard to overcome it, and he’s overcome some of it,” Alexander said at the time. “But not all of it.”

Schumer also interacts regularly with Republicans at the Senate gym. He prefers to work out early in the morning, when more Republicans than Democrats are likely to occupy the treadmills. That’s allowed him to develop relationships with Alexander, Sen. John McCain (Ariz.) and other Republicans.

For years, Schumer was known more as a gossip than an athlete in the members-only gym. But over the past two years, he said, he has developed a more rigorous weightlifting routine. He has lost 35 pounds, he said, while gaining a few new friends.