On July 17, a Vilnius court ruled to extradite the Lithuanian citizen to the US for prosecution.
The appeal will be heard by chairman of the Court of Appeals, Algimantas Valantinas, before handing down a final finding that is not subject to further appeals.
US law-enforcement officials accuse Rimašauskas, 48, of swindling Google and Facebook out of more than 120 million US dollars (EUR 103 mln).
Meanwhile, his lawyer wants Rimašauskas to be tried in Lithuania, where he is alleged to have committed the deeds he is accused of.
The Lithuanian is accused of sending fraudulent emails to the US companies, posing as employees of an Asian-registered company. It is suspected that he conned the Americans into wiring large amounts of money into Latvian and Cypriot bank accounts under his control. The man is accused of committing the crimes between 2013 and 2015.
US law-enforcement officials have brought fraud, money laundering and identity theft charges against the Lithuanian citizen. If convicted, the first two counts carry a maximum sentence of 20 years in prison.