"As per the directive received from the Government, data services are stopped at a few locations," Vodafone Idea's customer care tweeted.
According to the mobile carrier, text message services have also been suspended at a some locations.
Another mobile carrier, Bharti Airtel, said that it has suspended voice calls, text messaging and internet services in parts of the Indian capital.
Meanwhile, Delhi Chief Minister Arvind Kejriwal said that the government plans to use 11,000 free WiFi hotspots to cover the whole of New Delhi.
The Citizenship Act has earlier triggered Muslim student protests in India, with demonstrators believing that the new law violates the constitution by oppressing a group of citizens in a formally secular country on religious grounds.
Last week, the Indian parliament passed the bill, which allowed citizenship for Hindus, Buddhists, Jains, Christians, Sikhs and Parsis facing persecution in neighboring countries. Muslims were excluded from the right to citizenship under the bill, with the exception seen as a fresh discriminatory attempt to sideline the nearly 200-million-strong Muslim community.