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 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. JPMorgan, the biggest U.S. bank, said on Tuesday that its capital level will actually increase around 4% under the plan.

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. 

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 plan, 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.