A bill to overhaul the Dodd-Frank Act could get a vote in the full House as early as June 7.

The broad bill, sponsored by House Financial Services Committee Chairman Jeb Hensarling, R-Tex., would provide an “off-ramp” for banks that agree to hold a leverage ratio of at least 10% and would gut the Consumer Financial Protection Bureau, among other provisions.

One of the more politically charged provisions in the bill, a price cap on debit swipe fees will not be part of the legislation after Hensarling decided to remove it to increase the odds of the bill's passage in the House.

However, the Senate is unlikely to take up a similar package to unwind Dodd-Frank.

U.S. Rep. Jeb Hensarling (R-Texas)
The Financial Choice Act, authored by Rep. Jeb Hensarling, R-Tex., would gut the CFPB and allow banks to face simpler regulatory requirements in return for holding more capital.

"I would love to revisit Dodd-Frank. My community bankers have taken a hit,” Senate Majority Leader Mitch McConnell said last week during a TV interview with Bloomberg News. But the Kentucky Republican added that “there are enough Democrats to keep us from reforming Dodd-Frank” even though “all their community bankers are complaining about it.”

The House Rules Committee is scheduled to hold a hearing on Hensarling's Financial Choice Act on June 6 after Congress returns from its Memorial Day recess. A full floor vote could follow in short order.

It is unclear whether there will be amendments offered to make changes beyond a removal of the swipe fee provision. Democrats may also seek to use procedural tactics to slow the legislative process.

Democrats have also dubbed the legislation the “Wrong Choice Act,” seizing on a political opportunity to paint Republicans as doing Wall Street’s bidding.

However, most Wall Street banks do not strongly support the bill, and a Congressional Budget Office report found that most wouldn’t see much benefit if it became law.

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