According to CNN, travellers would also be asked to share their mobile phone contacts under the controversial proposals.
White House policy director Stephen Miller has already raised the idea with officials from the State Department, Customs and Border Patrol, and the Department of Homeland Security.
A White House source told the broadcaster that the policy was in its very early stages, and it is unclear how it would be implemented.
White House press secretary Sean Spicer is yet to have confirmed whether the policy is being looked into. The source said the Trump administration had discussed claims that San Bernardino terrorist Tashfeen Malik posted violent content on social media before he carried out the massacre.