NEW DELHI—Facebook Inc.’s WhatsApp is facing pressure in India to let authorities trace and read the encrypted messages of its more than 200 million Indian users in a new attempt at constraining global tech giants.

India’s telecommunications regulator has asked for feedback on new rules that—in the name of national security—could force “over the top” services such as WhatsApp, which use mobile operators’ infrastructure, to allow the government access to users’ messages.

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