Shepherd, 31, appeared in court on Tuesday when a judge decided against accelerating the process for his return to Britain, according to reports.
He could remain in Tbilisi, the Georgian capital, for several more months.
Before the hearing, a defence lawyer for Shepherd, Tariel Kakabadze, wrote on Facebook: “Prosecutor requested Jack Shepherd’s urgent extradition according to simplified rules.
“We, Jack Shepherd’s defense team are going to fight the request!”
Shepherd was convicted in his absence last year of killing Charlotte Brown, 24, on a first date in December 2015, when the speedboat they were in overturned in the Thames.
He handed himself into Georgian authorities last Wednesday.
The court appearance comes after the justice secretary said he was “very concerned” about death threats made against Shepherd’s lawyer.
David Gauke tweeted: “Very concerned about death threats made against Jack Shepherd’s lawyer. Such behaviour is completely unacceptable.”
Shepherd is being represented by Richard Egan, who has been sent death threats as he begins defending him.
Mr Egan has been sent a letter, which mentioned stabbing, bombing, the words “Heil Hitler” and a crude drawing of a swastika, according to the BBC.
The letter, which also stated that the sender had followed Mr Egan to his home, has been passed to the police.
Ms Brown died after the speedboat overturned during the night out in December 2015 – and Shepherd was found guilty of manslaughter by gross negligence and sentenced to six years in prison in his absence.
His disappearance ahead of his trial sparked an international manhunt and Ms Brown’s family had warned at the time that “justice is coming”.
The Crown Prosecution Service is working on a request to have Shepherd extradited to Britain to face justice over Ms Brown’s death.
The CPS said prosecutors were “consulting with the authorities in Georgia to progress our extradition request”.