A group of men approached the couple near a local railway station and demanded selfies with the woman before attacking them with stones on Sunday, the Times of India daily said.
It said the man suffered a fractured skull while his girlfriend, a 24-year-old Swiss national, had a broken arm and multiple bruises.
Bystanders took videos of the couple on their mobile phones as they lay on the ground after the attack, the paper reported.
Senior officer Amit Pathak told AFP by phone from Agra in the northern state of Uttar Pradesh that police had identified some of the attackers.
"Some of them could be minors, so we are investigating and if they are indeed minors they will be produced in juvenile court as per the law," said Pathak, senior police superintendent for Agra.
Another police officer, Pradeep Kumar, later said one suspect had been taken in for questioning.
The motive for the attack was not clear, but levels of sexual violence are relatively high in India and many Western countries warn visitors of the risk of attacks.
The Swiss embassy website warns tourists of high levels of crime in India, particularly in popular tourists spots such as Agra, where the country's most famous monument is located.
Indian Foreign Minister Sushma Swaraj said she had sought a report from the state government of Uttar Pradesh over the latest attack and would send representatives to visit the Swiss couple in hospital.
Some social media users criticised Uttar Pradesh Chief Minister Yogi Adityanath for failing to provide security to tourists.
"Such a shameful story on the breakdown of law & order in India's largest state. Is @myogiadityanath listening?" tweeted Milind Deora, a leader of the opposition Congress party.
Others said the incident was a blow to India's hopes of drawing more tourists to the country, which had just 8.8 million foreign visitors last year according to official data.