A senior police officer told BBC Hindi that four men attacked Quentin Jeremy Clerc and Marie Droz after an argument on Sunday in the northern city of Agra.
The incident came to light on Wednesday after local media reported that Mr Clerc had suffered a fractured skull and Ms Droz had fractures to her hand.
They are receiving treatment in a Delhi hospital. No arrests have been made.
Mr Clerc told The Times of India newspaper that the argument started after the men started taking pictures of them.
The report quoted police as saying they had forced Ms Droz to take selfies with them, and continued to follow them for an hour, before attacking them with sticks.
Agra's Superintendent of Police, Amit Pathak, said the suspects had been identified and would be arrested soon.
"The couple came to the police station and we took them to a hospital for treatment. They did not want to file a report, but we have still registered a case against four men," he said.
India's Foreign Minister Sushma Swaraj on Thursday asked the government in Uttar Pradesh state to submit a report in the case.