The push marks a political turning point for the big banks, which have largely kept a low profile in the wake of the financial crisis while they worked to regain favor with the public and with lawmakers. The few prominent attempts they made to raise their heads and influence policy quickly turned ugly. Citigroup, for example, was heavily criticized in 2014, when it became clear that the bank had heavily influenced language in a spending bill that would have rolled back certain post-crisis rules related to derivatives. Senator Elizabeth Warren, a Massachusetts Democrat, blamed the bank's sway over Republican lawmakers in the House of Representatives for nearly shutting down the government.
Now, less than four years later, the big banks are buoyed by a White House and Congress that sees deregulation as an overarching goal.
That has helped fuel efforts to relax some post-crisis rules that toughened oversight on all banks, regardless of their size and the risk their stumbles could pose to the financial system. The current push involves one such rule, which requires banks to hold a certain level of capital on their balance sheets based on their total asset size, regardless of how risky those assets are. For instance, cash or customer deposits held at the Federal Reserve are treated the same as riskier assets like subprime mortgages or junk bonds. The rule, known as the supplementary leverage ratio, is aimed at keeping banks from being able to take big risks without properly preparing for a disaster.
The Senate bill, sponsored by Mike Crapo, an Idaho Republican, would create a small exemption. As it is currently worded, it would most likely apply to just three banks, all of which take deposits primarily from large asset managers and other banks, rather than Main Street customers, and are known as custody banks.
The three custody banks — Bank of New York Mellon, State Street and Northern Trust — would, when calculating how much capital to hold on their balance sheets, be able to set aside deposits they received from other banks and immediately gave to the Federal Reserve or another central bank for safekeeping.
Jennifer Hendricks Sullivan, a spokeswoman for Bank of New York Mellon, said in an email that the exemption was structured in a way that "recognizes the unique business model of the custody banks."
Citigroup and JPMorgan argue that the exemption is not fair. They say that since they, too, take deposits from other banks and stash them at the Fed, they should get the same relief — even though that is not the primary focus of their business. Lobbyists for the two banks are hoping to persuade lawmakers to change the bill to allow all banks that accept custodial deposits to take advantage of the exemption, according to people familiar with the banks' efforts who spoke on condition of anonymity because they were not authorized to discuss those efforts.
"As Congress has sought to make a common sense change to the way capital rules treat custody assets, we have asked that they apply that change to all custody banks to maintain a level playing field in this important business," a Citigroup spokesman said in an email on Friday.
Spokesmen from JPMorgan and State Street declined to comment.
Whether the language is broadened or not, the presence of a carve-out has made supporters of the law's original capital requirement nervous. They say its purpose is to keep banks from convincing regulators that certain products aren't dangerous when they may ultimately pose a risk.
"This is not a tweak," said Sheila C. Bair, former chairwoman of the Federal Deposit Insurance Corporation. "This could be a very significant weakening of bank capital rules."
Ms. Bair and others who support keeping capital requirements unchanged for big banks point to lessons from the 2008 crisis that demonstrated how financial panics can change the risk profile of certain kinds of assets. For instance, before the crisis, many banks held securities that appeared safe and backed by highly rated mortgages, but they were actually backed by loans whose borrowers were at risk of default. Synthetic bonds and other similar products that received Triple-A ratings before 2008 became nearly worthless when the panic hit, bringing the country's financial system to the brink of collapse. The role of the leverage ratio enshrined in Dodd-Frank is to ensure that, next time, such miscalculations are impossible.
Aaron Klein, a fellow at the Brookings Institution who helped write Dodd-Frank as chief economist for the Senate Banking Committee, said changes to the leverage ratio could create a "slippery slope" in which banks start arguing for more exemptions, effectively blunting the rule's power. Banks could start arguing that other kinds of assets, like cash and United States Treasury securities, should not count.
"Once you start picking one thing or another to remove, where does it end?" Mr. Klein said.
The prospect of a lobbying fight is spooking smaller banks, which have been waiting years for a legislative vehicle that would relax restrictions on them, including requirements that they undergo regular stress tests. The bill's success hinges on the support of Democrats who may balk at voting for legislation that is seen as helping the industry's biggest players.
"It's everyone's right to lobby for whatever they want to lobby for, but this is a very careful, delicately balanced bill with very solid bipartisan support," said Paul Merski, the top lobbyist for the Independent Community Bankers of America, a trade association that has pushed hard for the changes central to Mr. Crapo's bill.
"If this should fail, then I don't think the financial sector will have an opportunity to get much relief brought up again in a long time," he said.
Even if the bill fails, the Fed could ease the burden on banks. Jerome H. Powell, the Fed chairman, said in congressional testimony last Tuesday that he favored changing the way the entire leverage ratio is calculated, which would naturally benefit the three custody banks, without making any special exceptions for them.
An aide to a Democratic senator who supports the legislation insisted that no changes would be made to benefit big banks like Citigroup. But members of the progressive wing of the Democratic Party have already been mounting their resistance to the entire bill.
Ms. Warren took to Twitter on Friday and warned that the bill, which she called the "bank lobbyist act," benefits only big banks and said she was planning a series of public events to decry the legislation.
And Senator Sherrod Brown, the Ohio Democrat who sits on the Senate Banking Committee, said in an interview that lawmakers who support the bill were suffering from "collective amnesia" about the financial crisis.
"Dozens of my colleagues think it's important for banks to make more money," Mr. Brown said. "The public is not calling for us to deregulate Wall Street."