Andrew Tate, 37, and Tristan, 35, were detained on Monday in Bucharest after Bedfordshire Police said it had obtained an arrest warrant.
The brothers can be extradited only after the conclusion of their separate trial in Romania on human trafficking charges, a judge ruled.
They have been released from custody.
Speaking outside court Andrew Tate said: “We are very innocent men and in time everybody is going to see that. We are excited to finish this judicial process and clear our names.”
Outside his house on the outskirts of Bucharest, he later told the BBC “the whole thing is garbage”.
The brothers, who “categorically reject all charges”, will remain under judicial controls in Romania which prevent them travelling outside the country.
Before the extradition was granted by the Court of Appeal in Bucharest, Bedfordshire Police confirmed it had obtained the arrest warrant for two men in their 30s as part of an investigation into allegations of rape and human trafficking.
“We understand this is a distressing subject, and people may be impacted by what they are seeing in the news,” the force said in a statement.
According to the brothers’ legal representatives the allegations date back to 2012-2015.
