Eveline Widmer-Schlumpf wants banks to go a step further than requiring a tax declaration from foreign customers, according to the Sonntags Zeitung.
She wants them to examine suspected cases of evasion to ensure they are respecting the law.
Sonntags Zeitung paper described the proposal as "explosive" as banks have so far fought against compulsory checks.
Widmer-Schlumpf, who as well as being finance minister is also the current Swiss president, will put the suggestion to ministers on Wednesday the newspaper said, citing several sources close to the finance department.
Widmer-Schlumpf is under pressure to reach an agreement with the United States over undeclared assets in Swiss accounts, as Switzerland tries to maintain the reputation of its banking industry.
US authorities are probing 11 Swiss banks as part of a clampdown on offshore tax evasion -- an investigation that recently prompted the collapse of Wegelin, the country's oldest private bank.
Three of its bankers have been charged with conspiring to help American clients hide funds.
In the coming months, Bern is expected to finalise a tax accord with Washington, similar to one already drawn up with Britain and Germany.
"We support the intentions of the finance minister," president of Widmer-Schlumpf's Conservative Democrats party, Hans Grunder, told the Sonntags Zeitung.
The head of the Socialists Christian Levrat said the suggestions "seem to go in the right direction."