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The Swiss government is considering a proposal to disclose bank client names and pay a multibillion-dollar fine to the United States to help resolve a long-running dispute between the two countries over the handling of tax-evasion cases, American and Swiss sources briefed on the matter said on Tuesday.

The fine, which could reach at least $7 billion to $10 billion according to these people, could be paid in part by the Swiss government, which would then seek reimbursement from the banks.

The possibility of an agreement is a turning point in a deepening conflict between Switzerland and the United States over the matter.

American prosecutors have been conducting criminal investigations of about a dozen Swiss and Swiss-style banks involving offshore private banking services that allowed at least tens of thousands of wealthy Americans to evade federal taxes. The Swiss government, which has long prized the secrecy of its banking system, now appears to be willing to cooperate with authorities, one person said.

The Swiss banks that have been the targets of investigations include Credit Suisse, which disclosed in July 2011 that it had received a letter saying it was under a grand jury investigation; the Zurich-based Julius Bär; two cantonal, or regional, banks; the Swiss operations of HSBC Holdings; and three Israeli banks, Hapoalim, Mizrahi-Tefahot Bank and Bank Leumi.

Members of the top echelon of the Swiss government known as the Federal Council are expected to discuss the matter as early as Wednesday, according to people briefed on the talks. These people asked to be unidentified because the discussions were continuing.

Hans Kaufmann, a Swiss parliamentarian and member of the conservative Swiss People’s Party, said that various levels of Parliament would be informed of the Federal Council’s decision in stages over the next 10 days or so.

“The Federal Council has already discussed the topic a few times in the last weeks, but has reached no conclusions so far,” said Anne Césard, a spokeswoman for the State Secretariat for International Financial Matters, a part of Switzerland’s finance ministry.

Eveline Widmer-Schlumpf, the head of the finance ministry, told Swiss radio on May 18 that “We are on the verge of presenting a solution,” adding that “it is clear that it will not be a pleasant solution.”

André Simonazzi, a vice chancellor and spokesman for the Federal Council, declined to comment until Wednesday on the matter. A spokeswoman for the United States Justice Department also declined to comment.

The bank Julius Bär acknowledged on Tuesday that it had received a formal request from American authorities for data on American clients of the bank’s offshore services. “We welcome the fact that the two governments obviously are close to an agreement on how to settle the tax dispute and will continue our early, proactive and cooperative steps addressing our situation,” said Sabine Jaenecke, a spokeswoman for Julius Bär.

American authorities have indicted dozens of Swiss bankers and their clients in recent years. A breakthrough came in 2009, after UBS, the largest Swiss bank, agreed to enter into a deferred-prosecution agreement. The bank turned over 4,450 client names and paid a $780 million fine after admitting to criminal wrongdoing in selling tax-evasion services to wealthy Americans.

In 2012, the Justice Department indicted Wegelin & Company, Switzerland’s oldest bank. The bank pleaded guilty in January, putting it out of business, and prosecutors have said off the record in recent months that more indictments could be coming.

Whistle-blowers have been a crucial part of the investigations and could continue to be so. Last year, the Internal Revenue Service said it would pay Bradley Birkenfeld, a former UBS banker, more than $104 million for his cooperation in revealing the secrets of the Swiss banking system and in that way helping to collect millions in unpaid taxes from wealthy Americans.

At least one other banker is said to be in discussions with American authorities. Swiss officials have been interested in a settlement since the United States threatened to press charges against cantonal banks. Most cantonal banks are wholly or largely owned by the cantons, or regions, in which they operate, making them state institutions backed by Swiss taxpayers.

Two cantonal banks, Basler Kantonalbank and Zürcher Kantonalbank, are under scrutiny by prosecutors in the Southern District of New York.

“Every bank under investigation negotiates on its own,” said Igor Moser, a spokesman for Zürcher Kantonalbank, in Zurich. He declined to elaborate but indicated that the bank would not be part of any global resolution that is reached.

Michael Buess, a spokesman for Basler Kantonalbank, said that the “negotiations with U.S. authorities has been defined as strictly confidential.”

In early May, prosecutors in New York sent a second request to the government of Liechtenstein, a tax haven now seeking to cooperate with the United States in rooting out tax evaders. Prosecutors are seeking details on eight or nine firms involved in setting up Liechtenstein-based foundations and establishments on or after Jan. 1, 2008, that may have helped American citizens evade taxes.

The request reflects efforts by prosecutors to “use Liechtenstein as a blunt wedge to pry open Switzerland,” according to a person briefed on the matter.