Bidve was visiting Salford in Manchester over Christmas holidays with other Indian students on Monday morning.
One 16-year-old is being questioned by the Greater Manchester Police along with a 17-year-old. The identity of the two teenagers has not been revealed.
He was with nine other Indian students when he was accosted by two white men, one of whom shot him from point blank range after a "very short" conversation.
Hailing from Pune, Bidve was a postgraduate student of Microelectronics at Lancaster University and was visiting friends in Manchester during the Christmas holidays with other Indian students when he was gunned down.
Meanwhile, a post-mortem revealed that Bidve, a postgraduate student of Microelectronics at Lancaster University, died as a result of "gun trauma" to the head.
The police have not confirmed what was said during the short conversation between the killer and his victim.
Detectives are continuing to make house-to-house inquiries and are reviewing CCTV footage.
Tributes to Bidve have been paid by friends and family on Facebook, where pages have been set up to help bring his body back to India, and also appeal for peace marches in memory of Bidve.
The police is not ruling out racism as a motive for the killing after the two men who approached the group of students early on Monday morning were described as white men.
Chief superintendent Kevin Mulligan said: "I have said previously I believe the answers as to who committed this awful murder lie in the community, and we have had an excellent response from people who share our abhorrence at what happened and want to see the killers brought to justice".
He added: "There is obviously speculation about why this young man was killed but at this stage it would be wrong to rule anything out or comment on that speculation while the investigation is in its infancy."
Bidve arrived in the UK in September for postgraduate
study at Lancaster University after completing an electronics degree at Pune University.
Bidve's father, Subhash, wrote in a message on his son's Facebook wall: "Please help in getting Anuj Bidve's dead body back to India. We appeal to all his friends in UK and India to help spread the word.”?
The family said it had tried to contact the Indian High Commission in London but the office was closed for Christmas.
Bidve's friends who were with him during the incident have given witness statements about his killing.
Without ruling out racism as a motive, Mulligan said: "This appears to be an unprovoked attack and we can't establish a motive at this time. We are investigating every possible aspect and we are in the early stages of the investigation, but we are not ruling anything out".
Initial witness statements have been taken from the other eight Indian students who were with Bidve.
They are now "in a safe place" being cared for by the police until they return to Lancaster university, Mulligan said.
A spokeswoman for Lancaster University said counseling and support has been organised for Bidve's friends who were described as "deeply upset".
London: The British police on Tuesday arrested two teenagers in connection with the unprovoked killing of 23-year-old-Indian student Anuj Bidve, who was shot dead yesterday in what could be a hate-crime.
First Published: Tuesday, December 27, 2011, 23:24