Last week, the lower house of the Swiss parliament voted against allowing the data to be passed on to the US, which alleges that almost 4,500 American citizens are avoiding tax by holding secret accounts with UBS.
While it has now reversed its decision and voted in favour of the plan, the lower house has also called for a referendum of Swiss citizens to take place on the issue.
But the idea of a ballot has been opposed by the upper house of the parliament, along with Switzerland's bankers' association, reports the Financial Times.
The referendum proposal is complicated by the fact that under Swiss law, citizens are granted 100 days to get the 50,000 signatures needed for a plebiscite.
If the full period was taken up, the August 19th deadline agreed between the US and the Swiss on delivering the customer information would be broken.
An agreement on what is to be done will have to be made between the two houses before the end of this week, when parliament is scheduled to break up.
Last week, a spokesman for the Inland Revenue Service warned: "We continue to monitor the events in Switzerland and we stand ready to pursue all legal options available to us should the Swiss fail to provide the required information."
The dispute has been rumbling for some time, with UBS agreeing to a $780 million settlement of criminal charges brought against it by US authorities in relation to aiding tax evasion back in February 2009.
A civil action was pursued against the bank in an attempt to access the names of the clients, with the Swiss agreeing to pass over the data in August last year.