Sources say that the police pressed for custody as they will require both the accused to conduct further investigation in relation to the complaint made by the victim.
Barala came under scrutiny after massive outrage after he was let off after initial arrest on Agust 5.
The development comes after the police earlier released a video footage showing the one of the accused (Ashish) buying alcohol before chasing Varnika Kundu's auto on the night of August 4.
Chandigarh police produced the accused at the Sector 43 district court here and sought their custody, as it was required to reconstruct the crime scene and verify facts.
Police said interrogation of the accused provided vital information, besides evidence from CCTV. However, he admitted that he was under the influence of alcohol.
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Earlier, the father of Vikas and BJP president Subhash Barala press conference on Wednesday for the first time on the matter.
The two accused were booked and re-arrested a couple of days ago under non-bailable sections of attempted abduction in the Varnika stalking case.
Taking out the "Bekhauf Azadi March" (freedom without fear march), the young and old said they were reclaiming the streets from "stalkers and elitist goons here". Subhash Barala's son and another person have been arrested.
Later, talking to reporters outside the court, the defence lawyer asserted that both Vikas and his friend did not commit any offence.
Haryana BJP chief Subhash Barala's son Vikas, accused of allegedly stalking and attempting to kidnap Varnika Kundu in Chandigarh, is not the only Barala family member to have been involved in a case of violence against women.