Serbian officials say Karadzic was captured in Belgrade on Monday, but Vujacic insists his client was captured last Friday on a public bus in a Belgrade suburb, then was hooded and transferred to an unknown location where he was kept for three days.
The lawsuit was reportedly filed against the alleged abductors.
"We have evidence and three witnesses who were there on the 18th [of July]," Vujacic said.
He also suggested that Karadzic could go on hunger strike and would refuse legal representation. He had previously said that he would launched an appeal against the extradition on Friday, the last possible day he can do so.
Karadzic faces 11 counts in The Hague, including genocide, for allegedly masterminding atrocities throughout the Bosnian war, including the Srebrenica massacre and the siege of Sarajevo.
Olga Kavran, the spokeswoman for the tribunal's prosecution, said on Thursday that she could not speculate about Karadzic's arrival.
"We're just waiting for the Belgrade authorities to inform us," she said.
Vladimir Vukchvich, Serbia's chief prosecutor for war crimes, told Al Jazeera in an exclusive interview that while Karadzic will appeal extradition, his arrival at The Hague is "inevitable".
"I don't see any reasons for hurry. If we waited for thirteen years to lay our hands on Karadzic from the day he was accused, I see no reason to hurry," he said.
"It may happen next week and I am certain it will happen but the final word is up to the court."
Serbian authorities continued to investigate who may have helped Karadzic assume the false identity that allowed him elude capture for more than a decade.
A spokesman for Serbia's war crimes prosecutor said those people would be found and prosecuted, adding that investigators were also trying to determine the true identity of Dragan Dabic, the name Karadzic used during his 12-year run from justice.
Several conflicting reports emerged about who Dabic really was, apparently because the name was common in Sarajevo at the time.
Karadzic's brother, Luka, went to visit his brother on Thursday at the courthouse where he is currently being held, but refused to disclose details of their encounter.
"I am his brother, I am not a lawyer ... I can't tell you anything," he told reporters.
But Karadzic's secret life is slowly being pieced together.
So far, details include a mistress, a bogus family he said he left in the United States, and regular visits to a bar called "The Madhouse" where he sipped wine below a picture of himself in his heyday.
|Karadzic's brother visited him in detention in Belgrade on Thursday [AFP]
Karadzic, who had been on the run for nearly a decade, had a girlfriend he presented as an associate in the alternative medicine business he ran, said a software engineer who says he was hired by Karadzic in February to set up a website.
Serbian newspapers on Thursday carried photographs of Karadzic's alleged girlfriend.
Karadzic introduced her only by her first name, Mila, who is in her early 40s.
He remained officially married to Ljiljana Zelen-Karadzic, who lives in their family house in the former Serbian stronghold of Pale, just east of Sarajevo.
Attempts to track down Mila for comment have been unsuccessful, but a neighbour described her as "a woman of virtue".
"The woman is very quiet and peaceful. She's a bit disappointed because she lost her job two years ago and she doesn't have money to live on.
"She's working on an ad hoc basis. She worked for 15 or 20 years. She's a woman of virtue," Urosha Maksimovic, who lives on the top floor of the building.
While most have welcomed Karadzic's arrest, it has also reawakened Serb nationalist feelings of those loyal to the former Bosnian Serb leader.
Dozens of ultra-nationalist demonstrators marched through the streets of Belgrade on Thursday protesting against his arrest, singing ultra-nationalist songs and chanting pro-Karadzic slogans.