Banks and foreign governments are mounting an increasingly desperate push against a sweeping US tax law that will force overseas institutions to report their American clients to the Internal Revenue Service.

The Foreign Account Tax Compliance Act was passed by Congress last year and comes into force in 2013. Last week, senior bank executives implored Tim Geithner, US Treasury secretary, to modify the law, according to people familiar with the meetings.

Banks say they are already racking up significant costs. Eventually, they say, the task of scouring records for US citizens and then reporting them could run into billions of dollars and conflict with domestic privacy laws. Disclosure records show groups including Switzerland’s Credit Suisse, Barclays of the UK and TD Bank of Canada have together spent millions of dollars lobbying on the issue.