They also appealed for a reduction of his 40-month sentence to reflect his contribution to the case.
But Judge William Zloch of the Federal District Court in Fort Lauderdale ordered the whistleblower to start his sentence on Friday as planned.
Former UBS banker Birkenfeld, 44, handed information to the US authorities to help with a tax probe into the bank.
The Swiss bank was targeted after allegations that it was helping US tax cheats to hide assets in UBS accounts.
Birkenfeld was sentenced by Zloch in August, just two days after US and Swiss authorities signed a pact in which Switzerland agreed to reveal the names of about 4,450 wealthy American clients of UBS to tax investigators. The whistleblower was widely seen as a key figure in the case. Justice Department officials said the jail time was justified because he was did not help the enquiry at first.