Washington (CNN) Democrats in Washington have begun discussing how to encourage Sen. Bernie Sanders to end his campaign without alienating his legions of supporters, as party leaders grow eager to unite the party behind Hillary Clinton and provide a more robust defense for her candidacy.

In private conversations on Capitol Hill, senior Democrats are weighing how to persuade Sanders to step aside without appearing as if they are trying to strong-arm him out of the race. In a phone call last month, Senate Minority Leader Harry Reid made the case to Sanders why it would make sense for him to leave the race after New Jersey and California vote on June 7, according to sources familiar with the conversation.

The widespread view, according to interviews with senators, House members and senior party officials, is that Sanders needs to see the writing on the wall himself: That he has no mathematical possibility to win the race and would be better-served to see his agenda enacted if he urged his backers to support Clinton.

"We will walk out of our convention with a nominee," Rep. Xavier Becerra, a California Democrat and active Clinton surrogate, told CNN. "We should be able to walk into the convention in a consolidating mode."

Some top Democrats privately say Clinton should consider the ultimate way to bring the progressive firebrand's supporters into the fold: Choose Massachusetts Sen. Elizabeth Warren as her vice presidential pick.

Senators and aides say they've felt reassured after recent discussions with Sanders and his advisers that he won't be a destructive force once voting concludes in mid-June. But Sanders has publicly vowed to take his fight to Philadelphia, something that could deprive the party of a critical month of healing and has spawned fears of unrest at the July nominating convention.

If he doesn't drop out, the options on how to persuade him to quit boil down to this: Propose potential process reforms, including gutting the role of superdelegates in choosing the next nominee, give him a prime speaking slot at the Democratic Convention and even dump the head of the Democratic National Committee, Rep. Debbie Wasserman Schultz, a controversial figure among Sanders' supporters.

So far, Sanders has not suggested publicly that his fight could be over after California, a state he has barnstormed in recent days as he tries to pull off an upset against Clinton.

"Our campaign has been dismissed and written off more times than I can count," Sanders said Wednesday in Palo Alto, California. "We're going to leave California with enormous momentum going into the convention. And I believe we've got a real shot to come out of that convention with the Democratic nomination for president of the United States."

Warren-Clinton talks pick up

Behind the scenes, discussions between the Warren and Clinton camps have been markedly increasing, especially as the freshman senator has begun to a play a more prominent role attacking Donald Trump, according to a source close to Warren.

And that has only spawned increased chatter that Clinton could pick Warren, who has not endorsed either candidate and is the only female Senate Democrat not to back Clinton.

Reid has spoken out publicly against Clinton choosing a senator as a running mate whose state has a Republican governor -- like Massachusetts -- fearing the vacancy would allow a Republican to be appointed and pad the GOP's majority.

But sources familiar with the situation tell CNN that Reid is open to the possibility of Warren as Clinton's running mate, in no small part because it would help unite the party after a long primary season.

"She can take away (Bernie's) power by showing there's no division within the party," said one Democratic source who is advocating Warren's selection.

It's uncertain when Warren would endorse Clinton, but the senator's allies believe she doesn't want to appear as if she's pushing Sanders to quit before he's ready to leave the race. Yet she also takes her role as a party uniter "seriously," according to the Warren source.

Watching Sanders closely

Much of how the party reacts after Tuesday's primaries will be dictated by Sanders' own actions, officials say. If he stays in the race, softens his rhetoric against Clinton and begins the process of healing the party, there will be less consternation. But if he blasts what he considers a rigged system, berates Clinton for her corporate donations and paid speeches to big banks on Wall Street and calls for a revolution at the convention, Democrats are bound to push back.

Sen. Chris Murphy of Connecticut, a Clinton backer, said his "preference" would be for Sanders to "take a hard look" after Tuesday and conclude he has no viable path to the nomination. He said he had no problem with Sanders continuing his campaign until July because he's confident that the party would be able to unite against Trump.

"One thing I'm slightly worried about is the tone and tenor of the convention," Murphy added. "We're going to need a very positive unified convention. He's going to have send very clear signals to his delegates that he wants them to be vocal and loud in support of Hillary in our convention."

Even if Sanders were to win California, where polls show the race with Clinton in a dead-heat, the math looks grim for him to win the 2,383 delegates to become the nominee. He would need 67% of the remaining pledged delegates to take the lead over Clinton in pledged delegates alone after the District of Columbia votes on June 14. Clinton's lead in superdelegates, however, means she is the only candidate who can secure enough delegates to win the nomination before the convention.

Reid, in an interview with the Associated Press Wednesday, made his strongest comments yet about Sanders' bleak chances, saying the "math is the math" and that "sometimes you just have to give up."

"I think he better do a little mathing," Reid said.

The comments are significant because Reid -- who backs Clinton -- has avoided criticizing Sanders through the course of the campaign. Reid is perhaps Sanders' closest friend in the Senate; the two had an emotional call when Reid informed Sanders he was backing Clinton in February. And Sanders' wife, Jane, hugged Reid and thanked him for staying neutral before the Nevada caucuses as well.

But Reid and Democrats know they need the support of Sanders to help bring his dedicated backers behind Clinton. After the Nevada convention last month spawned angry protests from Sanders' supporters, many argue that the senator has to play a major role to unite the party -- or risk electing Trump. Reid has made the case repeatedly to Sanders, and others have publicly as well.

"My sense is if he believes as passionately about the issues he's been talking about on the campaign trail," Becerra said, "then, after June 7, there's a passionate need for him to be advocating for someone in the White House who will make much of what he is talking about a reality."

Murphy also was optimistic that Sanders would exit gracefully.

"He's going to be an incredibly impactful spokesman for Hillary," Murphy said. "Once the president and Elizabeth Warren and Bernie get behind Hillary, that's going to be impactful. I'm not wringing my hands on whether it happens next week or next month."