Firtash denies any wrongdoing.
The Austria Press Agency reported that Justice Minister Clemens Jabloner has approved the extradition. But a Vienna state court judge ruled it could only take place after that court has decided on the defense motion to reconsider the matter, and court spokeswoman Christina Salzborn said the defense provided "extremely extensive material."
Dieter Boehmdorfer, an Austrian lawyer for Firtash, told Oe1 radio the defense aims to prove that Firtash committed no crime in India and there is no suspicion against him there. It also aims to prove "that the U.S. does have a far-reaching political motivation" to secure Firtash, he added.
Firtash was arrested in Austria in 2014 and then freed on 125 million euros ($141 million) bail. In a protracted legal battle, a Vienna court initially ruled against extradition on the grounds that the indictment was politically motivated.
A higher Vienna court in February 2017 rejected that reasoning as "insufficiently substantiated" and ruled that Firtash could be extradited. Austria's Supreme Court of Justice upheld that ruling last month.
Also in June, a Chicago federal judge rejected a motion to dismiss the indictment against Firtash, who has argued that the U.S. has no jurisdiction over crimes in India. However, federal Judge Rebecca Pallmeyer ruled that it does, because any scheme would have impacted a Chicago-based company.
American aviation giant Boeing, based in Chicago, has said it considered business with Firtash but never followed through. It is not accused of any wrongdoing.
Before becoming Trump's campaign manager, Manafort pursued a business deal with Firtash to redevelop a Manhattan hotel. Manafort isn't accused of wrongdoing in the Chicago case.
Manafort was sentenced in March to more than seven years in prison on federal charges in cases from special counsel Robert Mueller's Russia investigation. The charges related to Manafort's years of Ukrainian political consulting work. Manafort also faces fraud charges in New York state.