WASHINGTON, April 17 (Reuters) - The U.S. Federal Reserve's vice chair for supervision, Michelle Bowman, has told big bank executives that she does not expect the industry to stage another aggressive pushback in a bid to win further capital relief, according to three people with knowledge of the communications.

The Fed last month unveiled relaxed new drafts of the "Basel III" and "GSIB surcharge" rules which the central bank estimated would reduce capital levels at big U.S. banks by around 4.8%, in a victory for the industry which fought fiercely to water down the Fed's original 2023 plan that had envisaged 20% hikes.

Still, the benefits will be unevenly distributed, and not everyone is happy, according to the sources. In a surprise twist, JPMorgan, the biggest U.S. bank, said on Tuesday that its capital level will actually increase by around 4% under the plan. In his annual shareholder letter this month, the bank's CEO, Jamie Dimon, said the proposals were still "very flawed" and that aspects were "un-American."

Other big bank executives said during earnings this week that the industry would likely seek some changes and expected to feed back to the Fed during its formal 90-day comment period. 

Big banks fought the 2023 capital plan with an  unprecedented campaign that included lobbying congressional lawmakers, billboard posters in the District of Columbia area, prime-time commercials and threats to sue. They argued that it would stifle lending and harm the economy. That plan was led by Bowman's Democratic predecessor, Michael Barr. 

BILLBOARD BATTLE OVER CAPITAL RULES

Bowman and other Fed officials have communicated in meetings with bank executives in recent weeks that they have worked hard to address bank complaints and do not expect the industry to reprise the aggressive tactics they deployed when fighting the 2023 proposals, the three sources said.

Officials have conveyed that industry comments, which are due by around mid-June, should be limited and specific, two of the people said.

A spokesperson for the Fed declined to comment.

The Fed's message poses challenges for JPMorgan and other lenders, which feel they have lost out or gained less than their peers, and have a limited opportunity to lock in favorable changes. 

It also underscores Bowman's eagerness to finalize the rules after a vicious, years-long battle that became an embarrassment for Fed Chair Jerome Powell and Barr.

Speaking at the Institute of International Finance conference in Washington this week, Bowman signaled she wanted to move fast, saying she would take on feedback but still hoped to finalize the rules this year. 

"I'm sure not all of it is going to be positive, but my hope is we've struck the right balance, I think it's a very middle-of- the-road, reasonable proposal," she said. 

BOWMAN'S BALANCING ACT ON FED BOARD

Though critics have warned that looser capital rules could create hazards for the financial system, Bowman, an appointee of President Donald Trump, and Treasury Secretary Scott Bessent have argued that economic growth will bolster financial stability, and that the changes will be more sensitive to real risks.

While officials are keen to put the Basel fight behind them, November's midterm elections are another reason to move fast, say other executives and analysts. Those elections could hand one or both chambers of Congress to Democrats, who may ramp up scrutiny of the proposals which some have characterized as a Wall Street giveaway. 

Giving banks further wins, meanwhile, could also alienate two of the Fed board's Democratic governors, Lisa Cook and Vice Chair Philip Jefferson, who voted for Bowman's proposals. Barr, who stepped down from the Supervision post last year but remained on the board, dissented.

With Republicans in the majority on the board, Bowman does not need their votes, but the central bank has historically strived for consensus.

"We worked very hard to have as much consensus as possible," said Bowman at the conference.