Visa and banks will press lawmakers on Thursday for relief from a proposed slashing of debit card processing fees, an issue that has attracted bipartisan support for softening the impact of part of the Dodd-Frank financial law.

Banks and card network companies such as Visa Inc and MasterCard Inc have been pleading with the Federal Reserve to change a December proposal that would cut debit processing fee revenue by about 75 percent.

But even debit industry sources admit that getting the Fed to significantly ease up on its proposal will be a tough task, leading the industry to focus its sizable lobbying firepower at Capitol Hill.

A House Financial Services subcommittee on Thursday will hear from Fed Governor Sarah Raskin and industry representatives on the issue.

The Dodd-Frank law passed last year called for debit card fees that are reasonable and proportionate to the issuers' costs, a provision added with the strong support of retail industry groups.

The Fed is charged with implementing that provision and in December proposed capping the so-called interchange fees at 12 cents per debit transaction -- nearly a 75 percent cut from the 2009 average of 44 cents per transaction.

The proposed caps would shave some $13 billion off the industry's annual $23 billion in debit card processing revenue, according to CardHub.com, and could also terminate some of Visa and MasterCard's exclusive processing deals with banks.

Card industry lobbyists believe they have a receptive audience among Republicans who now control the House of Representatives.

Visa Inc General Counsel Joshua Floum told Reuters on Wednesday that some lawmakers are close to proposing legislation to delay the implementation of the rule, the final version of which is due in April.

We are optimistic that something will come soon given our discussions with various members of Congress, Floum said, adding that Visa has reason to believe that there is a lot of sentiment to suspend and study the law.

No lawmaker has stepped forward with a bill, however, and even if the House were to pass legislation it would face stiff resistance in the Senate.

Senator Richard Durbin authored the provision, and as a member of the Senate's Democratic leadership he is in a good position to bat away any challenges to the fee restrictions.

In the House, at least, there is some bipartisan support for easing the fee restriction.

Democratic Representative Barney Frank, a co-author of the Wall Street reform law, said he supports making some changes.

In an interview on Wednesday he said Republicans have approached him about what type of bill he would be open to supporting.

Frank said he has told them he does not support delaying the fee restrictions implementation but could support instructing the Fed to include more factors in determining what is a reasonable amount a bank can charge a retailer when a debit card is used.

(Reporting by Dave Clarke in Washington and Maria Aspan in New York; Editing by Tim Dobbyn)