Indian police made fresh arrests in the case of the execution of a Hindu tailor in Rajasthan, a murder that stoked unease between the Hindu majority and Muslim minority, leading to a clampdown on protests and the internet to prevent tensions from escalating.

Three senior police officials said on Saturday that two Muslim men based in the northwestern state were held for allegedly planning the tailor's murder last week in his shop in Udaipur, a popular tourist destination dotted with lakes and palace hotels.

"We have now arrested the two masterminds and previously we had arrested two men who committed the heinous crime," said Prafulla Kumar, a senior police official based in Udaipur.

Kumar said internet services were being gradually restored and security forces continued to be on alert following the murder, allegedly carried out by two Muslim men now under arrest who filmed the act and posted it online.

The alleged perpetrators said the act was in response to the victim's support for a BJP politician Nupur Sharma's derogatory remarks about Prophet Muhammad (Peace Be Upon Him).

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The victim, Kanhaiya Lal Teli, had allegedly put up a social media post supporting the former spokesperson for Prime Minister Narendra Modi's party who made anti-Islam comments in May.

On Friday, judges from the Supreme Court of India stated Sharma must apologise to the whole nation after the remarks intensified religious fault lines in India, angered Muslim nations and triggered diplomatic strains.

In India, at least two demonstrators were killed in police fire during protests against Sharma's comments.

Read: Houses razed as India steps up crackdown to stop unrest

In Afghanistan, militant group Islamic State last month claimed an attack on a Sikh temple that killed at least two people and injured seven was in response to insults levelled at Prophet Muhammad (PBUH) in India.

Police in New Delhi arrested journalist Mohammed Zubair, a vocal critic of the Modi government, who had helped draw attention to Sharma's remarks through his fact-checking website Alt News and on social media.

The National Investigative Agency (NIA) — India's top anti-terrorism agency — said they were carrying out a probe in the Hindu tailor's killing.

A senior NIA official in New Delhi said they were questioning Muslims linked with the four suspects in Udaipur to identify whether they had links with militant networks.

Muslims living about 3 kilometres from the tailor's shop where the victim was killed said they felt nervous and feared a social and economic boycott by powerful Hindus residing in Udaipur.

"I know what has been done is barbaric but the community should not be held responsible for the deed of two people," said Mohammad Farukh, a medical representative living in a Muslim-dominated area of the city.

The All India Muslim Personal Law Board called the incident "highly condemnable", adding that it was against both Indian law and Islamic strictures.