By John Whitesides and Amanda Becker

WASHINGTON (Reuters) - Winning over Bernie Sanders supporters who flocked to his insurgent presidential campaign will be the first and possibly toughest order of business for Hillary Clinton after she became the presumptive Democratic presidential nominee this week.

Every time the Vermont senator mentioned Clinton's name at a rally in California on Thursday night, his supporters booed as if she was Donald Trump, the Republican businessman who will face off against Clinton in the Nov. 8 election.

But Clinton is wasting no time in launching her charm offensive. She called Sanders on Tuesday night, and during a celebratory rally in Brooklyn she praised his candidacy, saying their "vigorous debate" during the nominating contest had been good for the Democratic Party.

Democrats in both camps said they expect to see more specific olive branches offered to Sanders in the weeks ahead, including compromises on the issues platform to be adopted at the party's July nominating convention and reforms in a Democratic primary process that Sanders criticized as rigged to favor the establishment.

It may not be an easy sell for all of Sander's supporters, who have helped catapult him from political obscurity into the national spotlight, cheering his message on income inequality, campaign finance reform and Wall Street excesses. Given little chance a year ago, Sanders earned nearly 10 million votes and won more than 20 states during the nominating process.

A Reuters/Ipsos poll in May found Sanders supporters have become increasingly resistant to Clinton in the past few months, with less than half saying they will vote for her if she becomes the party’s nominee.

Last month, 41 percent of Sanders' supporters said they would vote for Clinton over Trump in the general election. But that was down from 50 percent in April, and 52 percent in March.

And some groups supporting Sanders plan to ramp up the pressure on superdelegates, prominent party officials who can vote for any candidate, in the last few weeks before Clinton’s nomination becomes official at the party convention in Philadelphia.

The people behind the Occupy DNC Convention Facebook group, with around 25,000 members, are creating their own web-based app to contact superdelegates, planning daily emails to urge them to switch to Sanders.

CLINTON'S TRUMP CARD

Clinton moved to the left to fend off Sanders during the primary battle, opposing the Trans-Pacific Partnership, an Asian trade pact, and the Keystone oil pipeline from Canada, voicing support for a higher minimum wage and backing Sanders' general goals of reining in Wall Street and reducing income inequality.

Her best move now to win over progressive voters who backed Sanders would be showing she will stand behind those positions.

"One of the biggest points of skepticism that we hear from Sanders supporters and even lukewarm Clinton supporters is the question of whether she really meant the stuff she said in the primary," said Adam Green, co-founder of the Progressive Change Campaign Committee.

"The next two weeks are a golden opportunity to make the answer apparent," he said, citing her pledges to seek criminal sentences for financial fraud, break up banks that pose economic risks and expand Social Security benefits.

Several Democrats said Trump may prove to be the chief motivator for Sanders supporters to switch to Clinton.

"Trump has done more than anyone to push the Sanders people toward Clinton for the general election," said Brad Anderson, an Iowa-based Democratic strategist who supported Clinton in the primary and said he has seen signs of a shift in the last few weeks.

Clinton could make specific concessions or overtures to assuage Sanders, particularly by eliminating the superdelegates from the primary process. Sanders has complained about the involvement of superdelegates, who have heavily backed Clinton.

Another easy way to get Sanders' most passionate supporters on board is to simply hire them, says Democratic strategist Steve Schale.

That's what Schale, a Florida-based staffer for President Barack Obama's 2008 presidential campaign, did once the bitter primary fight between Obama and Clinton ended that year. Florida was crucial to Obama’s prospects — and his campaign need a large operation to work on voter registration, persuasion, and turnout.

Clinton could offer Sanders diehards a similar opportunity to stay involved. "For a lot of these folks, the campaign is as much as social calling as it is a movement," he said.

While there were concerns in 2008 that Clinton's supporters would be too disaffected to support Obama, Clinton endorsed him less than a month after the primary ended and unified the party.

Democratic party leaders may also help ease tensions. Sanders will meet on Thursday with Obama and Senate Democratic leader Harry Reid of Nevada.

(Additional reporting by Luciana Lopez and Chris Kahn in New York, Ginger Gibson in Washington, James Oliphant in California)