The Swiss daily Neue Zuercher Zeitung, quoting sources familiar with the talks, said that the fine could reach 40 percent of the funds in question.
Swiss authorities have been tightlipped about the content of the accord, which is still being finalised, but Finance Minister Eveline Widmer Schlumpf said Saturday that the conclusion of the negotiations was in sight.
US authorities have repeatedly accused Swiss banks of complicity in tax evasion, because they hold deposits worth billions of dollars belonging to American citizens that have not been declared to the tax authorities.
"The banks won't get this for free. It's clear that it won't be a pleasant solution," Widmer Schlumpf said Saturday, without giving details.
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Swiss media have reported that under a compromise hammered out by negotiators, the 300 banks in Switzerland would be classed according to their level of alleged complicity in tax evasion.
The dozen banks seen as the main perpetrators would reportedly be forced to make a case-by-case deal with the United States.
A second category, comprising those with American clients but which have not yet faced legal action in the United States, would have to pay a fine.
A third group would include banks with just a few US clients.
Besides taking on US tax dodgers and Swiss banks directly, Washington also succeed in April 2012 in pressuring the banks to hand over the names of 10,000 employees with American clients.