Sitting together is a start, but a depleted group of Senate moderates is looking for more concrete ways to rebuild relationships across the aisle — including more bipartisan lunches, a Camp David summit and maybe even getting rid of the aisle altogether.

Looking to advance the bipartisan seat-sharing for the State of the Union address further, Sen. Ben Nelson said Tuesday that he wants to do away with partisan seating permanently as part of an effort to renew cross-party coordination.

“I’m going to advocate … if there is truly an interest in working together, that we eliminate the aisle on the floor of the Senate,” the Nebraska Democrat said. “Sit not by party but by state or by alphabet or in some other way. But it will only work if it’s more than just symbolism.”

And that gets to the hard part — while moderates should have more power in a closely divided Senate, their ranks were hit hard by retirements and defeats in 2010.

Last week’s retirement announcements from Homeland Security and Governmental Affairs Chairman Joe Lieberman (I-Conn.) and Budget Chairman Kent Conrad (D-N.D.) are the latest blows to centrist ranks.

The remaining moderates have to wrestle the desire for bipartisan accomplishments with the cautionary tales of politicians such as Lieberman, who was defeated in the 2006 Democratic primary, or former Utah Sen. Bob Bennett, who lost a GOP state convention fight.

But that doesn’t seem to be stopping a flood of bipartisan proposals from emerging as the reality of a closely divided Senate sets in.

Bipartisan bills on new budget rules, various deficit reduction efforts and even under-the-radar issues such as patent reform are gaining steam.

Republican Conference Chairman Lamar Alexander said there is a scramble under way in both parties to find bipartisan legislation.

“The recognition is that in an evenly divided Senate, which we’re likely to have for several years, nothing happens if you don’t work across party lines, and once that dawns on people, we are going to see a lot more consensus on major issues, and we can start with the debt,” the Tennessee lawmaker said.

But Alexander, who hosts bipartisan breakfasts with Lieberman and receptions with Sen. Charles Schumer (D-N.Y.), said more interaction between the parties has to be part of the solution.

“Senators spend almost all our spare time in team meetings during which we talk about what we’re going to do to each other,” he said. “We’d like to have some other occasions to get together where we can talk about what we might work on together. The bipartisan breakfasts are one; the receptions are another. There’s just too few of them.”

Sen. Mark Udall told reporters to “stay tuned” on future proposals aimed at fostering bipartisanship.

“I think we all agree that if we can’t sit together at an important night like this, how can we face the real challenges that the country has: how we’re going to pay down the debt, fix a broken immigration system or develop a 21st-century energy policy ­— much less get our economy back on its feet and focus on jobs as job one if we can’t sit together,” the Colorado Democrat asked.

Sen. Claire McCaskill, who is sponsoring various budget reform bills with Republicans, including spending limitation bills with Sens. Bob Corker (Tenn.) and Jeff Sessions (Ala.), said she has been pushing for bipartisan seating since she was elected.

“Sitting together can’t hurt, and it might just help,” the Missouri Democrat said.

Lieberman took her up on it in the Homeland Security panel, where Democrats sit alongside Republicans.

“I was bipartisan before bipartisan was cool,” she quipped.

Moderate Sen. Tom Carper said he’s pushing other efforts to bring the parties together behind the scenes. However, similar moves have been made each of the past several Congresses but tend to lack staying power, the Delaware Democrat said. “It starts strong and generally fades away before the first year is over,” he said.

Carper echoed Alexander’s sentiments about the lack of time Members of both parties spend interacting during what has grown to be more limited time spent in Washington, D.C., over the years.

Senators should have bipartisan lunches at least six times a year, Carper said. He has also urged President Barack Obama and much of his Cabinet to schedule an informal summit at Camp David to bring the leaders of both parties and their spouses together for a weekend.

Carper said that Obama is moving to the center and that centrists should help him triangulate against the extremes in both parties, like they did in the 1990s when President Bill Clinton had to work with a Republican Congress.

“The environment that faces us and the new Congress is one that should give centrists in both parties an opportunity to help the administration to try and move to the center,” he said.

Carper said there are several areas that are tailor-made for the new Congress, including education reform.

“It could be a confidence builder … rebuilding some trust and build on what happened during the lame duck,” he said.

Meanwhile, Sen. Mary Landrieu said the talk of the demise of moderates is overblown. A new crop has emerged to take the place of those who have departed, the Louisiana Democrat said.

“We’re sorry to see Sen. Lieberman, Conrad, [Evan] Bayh and [Blanche] Lincoln go, but these ranks have been replenished with people like Jim Webb, Mark Warner, Kay Hagan,” Landrieu said.

And the success of various efforts to rebuild bipartisanship will rest largely with Republicans, she said.

“What we are looking for is moderates on the Republican side,” Landrieu said, adding that it’s “frightening” how few Republicans were willing to compromise in the last Congress.

“The next move really is on the Republican side,” she said. “Over a dozen Democrats have shown a willingness to compromise. The next move is with John McCain, Lindsey Graham, Scott Brown. … We can’t just keep doing all of the moving to the center.”

McCain, for his part, said there is room for moderation in the Senate, pointing to the growth of independent voters in his state of Arizona and across the country. He’s working with Carper on an enhanced rescissions bill that is a less powerful version of a line-item veto.

Carper, however, acknowledged Republicans are in a tough spot, noting then-Rep. Mike Castle’s defeat in his state’s Senate GOP primary last year, as well as Bennett’s defeat in Utah.

“Republicans are between a rock and a hard place. … Very good, conservative but centrist Republicans were punished for their willingness to work across the aisle. It’s a tough call for Republicans.”