WASHINGTON—Cryptocurrency companies want to tap into the Federal Reserve payments systems that traditional banks use to move money around quickly. The banks are pushing back.

The companies include Avanti Bank, which aims to provide custody services for institutional investors in cryptocurrencies, and Kraken, a cryptocurrency exchange platform. They say direct access to the Fed’s payment systems would allow them to more quickly and cheaply process orders from customers buying and selling digital assets. Currently they must partner with traditional banks that have accounts with the Fed.

Traditional banks say the newer financial firms are supervised relatively lightly and lack the internal controls needed to ensure against money laundering and other illicit activities—concerns that regulators have expressed about the crypto industry more broadly. And they say the firms are riskier because they aren’t insured by the Federal Deposit Insurance Corp.

“It is reasonable to expect that such applicants will pose heightened risks regarding matters of anti-money-laundering, cybersecurity and consumer protection, as well as safety and soundness,” the Bank Policy Institute, which represents large banks, and the Independent Community Bankers of America wrote in a letter to the Fed last month.

Avanti and Kraken, which both have “special purpose” bank charters in Wyoming, say they have all the same compliance, controls and supervisory requirements of traditional banks. The only U.S. bank regulator that has a supervisory exam manual for crypto is in Wyoming, they say.

If they have their way on access to the Fed’s payment systems, that could encourage more firms to follow their example, introducing more competition for banks.

“It has the potential to reduce banks’ traditional role as gatekeepers and toll collectors for payment flows that are likely to grow over time,” said Jonah Crane, a partner at Klaros Group, an advisory and investment firm.

Last year, the central bank processed about $900 trillion in payments on its systems. These ranged from small bank-to-bank payments such as direct deposits or automatic bill payments to large wire transfers between financial institutions.

The struggle over access to the Fed’s payment systems also reflects incumbent banks’ concerns about the potential for competition from larger tech companies, such as Facebook Inc. and Google parent Alphabet Inc., which don’t face the same level of federal bank regulation.

The Federal Reserve is trying to figure out how to keep cash relevant in a cashless world. It is considering digitizing the U.S. dollar, giving people money they can access on their phones and bypassing electronic payments that can be slow and costly for businesses. Illustration: Jacob Reynolds/WSJ The Wall Street Journal Interactive Edition

“They have some reason to be paranoid,” said Eugene Ludwig, a former comptroller of the currency, who was responsible for the regulation of large national banks.

For example, Diem Association, which is backed by Facebook and 25 other members, has said it is developing a blockchain-based payment network that will be faster and cheaper than existing systems while protecting consumers and providing safeguards against financial crime.

Diem is partnering with a Fed-regulated bank on the project. But if other tech companies got direct access to the Fed’s payment systems, they may not need to take the additional step of partnering with established banks, according to banking lawyers and former regulators.

Regulators are also concerned that some types of crypto activities could pose risks to financial stability if they grow big enough. For example, some officials worry that so-called stablecoins—a form of digital currency pegged to the value of the dollar and other traditional currencies—could be susceptible to the kinds of runs that affect banks and mutual funds in a crisis.

Fed regulators are seeking feedback from the industry on a set of proposed principles for regional Fed banks to consider when evaluating applications for access to the payment systems from nontraditional banks. Among them: whether strains at the institution may be transmitted to other segments of the financial system in times of stress.

The Fed suggested that federally insured banks already are likely in compliance with the guidelines. Firms that aren’t “may require more extensive due diligence,” the Fed said.

In recent years, startup financial firms have benefited from the issuance of new types of charters at the state level, and approval to operate as trust banks at the federal level. While full-service banks typically engage in three core activities—deposit taking, lending and payments—these nontraditional institutions offer only some of these services. They generally aren’t allowed to lend to customers, but some are authorized to accept deposits while others collect fees for maintaining custody of the private digital keys for crypto customers or operate crypto exchanges.

Caitlin Long, chief executive officer of Avanti, said granting direct access to the Fed’s payment systems to banks that cater to the digital asset sector should be welcomed because doing so would bring them under the watchful eyes of regulators.

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“The absence of action to open a direct path has pushed much of the U.S. dollar banking of the digital asset industry into the ‘shadow’ banking system, which means risks cannot be readily monitored,” she wrote in a July comment letter.

Still, after about a year of waiting for Fed approval, Ms. Long said in an interview that her firm this month applied to be directly regulated by the central bank, a move she said should bolster its case for access to the central bank’s payment systems. If approved, her bank would also be directly subject to Fed regulation and examination.

Kraken also sought direct access to the Fed’s payment system last year and is still awaiting an answer.

The cryptocurrency exchange operator describes itself as just as safe as an FDIC-insured bank because it doesn’t lend out its depositors’ money and holds 100% of their cash at a correspondent bank or at the Fed, via its correspondent institution. It is also supervised by the Wyoming Division of Banking.

“I agree these banks need to have a bank-grade supervisory and oversight program,” said David Kinitsky, chief executive of Kraken Bank, a wholly owned subsidiary of Kraken. “We do.”

Write to Andrew Ackerman at andrew.ackerman@wsj.com