The Swiss daily Neue Zuercher Zeitung, quoting sources familiar with the talks, said that the fines could reach 40 percent of the funds in question.
Swiss authorities have been tightlipped about the content of the accord, which is still being finalized, but Finance Minister Eveline Widmer-Schlumpf said on 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.
Starting with accusations against a dozen banks, the dispute ballooned and Switzerland launched talks to try to reach an overarching settlement and provide legal closure for its financial sector.
"The banks won't get this for free," Widmer-Schlumpf said, without giving details.
"It's clear that it won't be a pleasant solution."
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.
This group reportedly includes Credit Suisse, Julius Bär, Wegelin, ZKB, BKB, Pictet and Neue Zürcher Bank, as well as the Swiss arms of Britain's HSBC, Liechtenstein's LLB and Israel's Leumi, Hapoalim and Mizrahi.
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.
The banks gave the names to US authorities after a green light from the Swiss government, which has faced criticism from banking sector employees for potentially exposing them to charges of abetting tax evasion.
Switzerland's long-held banking secrecy rules have gradually been weakened over recent years as crisis-struck countries press the Alpine nation to reveal details of their citizens who place their assets here.
Any deal would require not only Swiss government approval, but also the backing of parliament.