Thirty-one-year-old Mahtab Ahad Savoji's naked body was found in suitcase in a canal near the Venice Lido in northeast Italy, reports said Monday.
Investigators believe Singh and Kaur strangled the set design student on 27 January at the flat they shared with her in the northern Italian business capital Milan, put her corpse in the suitcase and took it by train to Venice, a city where Singh had worked.
Police recovered the suitcase with Savoji's body inside on 28 January.
The Indian couple was arrested on Monday after CCTV footage showed them arriving at Venice's Santa Luca station station with the suitcase and later at the Lido's landing stage, but they were not initially named in reports.
Singh and Kaur said they had found Savoji dead after a drinking binge, panicked and decided to dispose of her body. But their claims were disproved by marks on her corpse and incriminating mobile phone conversations between the couple, according to reports.
Savoji's relations with the Indians had soured and the couple may have killed Savoji after she said she wanted to move out and asked them to give her back her share of the deposit, some reports said.
One reported line of enquiry is that Singh had taken a fancy to Savoji and that she had decided to go and stay with a girlfriend after receiving unwelcome sexual advances from the Indian.
After dumping the suitcase containing Savoji's body in the canal on 28 January, Singh and Kaur paid 500 euros for a taxi to drive them back to Milan, reports said.
The two suspects face charges of murder and concealing a corpse.
Police from Milan and Venice are cooperating in the investigation, which is being coordinated by prosecutors in Milan.