Choksi, in an affidavit earlier this week, had claimed that he was unable to return to India due to persistent health problems.
Choksi and his nephew Nirav Modi are wanted by the ED and the CBI for allegedly defrauding the PNB to the tune of Rs 13,400 crore in collusion with a few employees of the government-run lender.
Judge Ingrid Simler at the Royal Courts of Justice said that there are substantial grounds to believe that Modi will fail to surrender.
UK's Westminster Magistrate's Court last month denied bail to Modi, who has been behind bars at one of England's most overcrowded prisons, Her Majesty's Prison Wandsworth.
'Mr Modi can appeal to the High Court as of right. He does not need leave to appeal,' a spokesperson said.
The court extended the fugitive diamond merchant's next remand hearing for June 27.
The 48-year-old diamond merchant has been lodged at Wandsworth prison in south-west London since his third attempt at seeking bail was rejected.
In a reply to an RTI query, it said requests for extradition of Mallya and Modi have been sent to the UK government.
The 48-year-old is back in prison after his bail was denied on Wednesday by Westminster Magistrates' Court Chief Magistrate Emma Arbuthnot.
While his two previous bail pleas have been rejected, he can make a third application if there is a considerable change in circumstances.
13 vehicles, 11 belonging to Nirav Modi group and 2 belonging to Mehul Choksi group, were put on e-auction through MSTC platform on April 25.
On Thursday, ED auctioned thirteen luxury cars of Nirav Modi and his uncle, Mehul Choksi.
The hearing follows his bail being rejected by Chief Magistrate Emma Arbuthnot on March 29 on the grounds that there was a "substantial risk he would fail to surrender".
Of these 13 vehicles, the BMW and Toyota Innova Crysta belong to Choksi and the rest belong to Modi, his family and his group companies.
An ED special director, based in Mumbai, heads the western region of the agency and has control over Maharashtra, Gujarat, Madhya Pradesh and Chhattisgarh.