The move, which the White House official cautioned was far from becoming official policy, would likely be controversial among diplomats and experts in State Department matters.
The bureau of Consular Affairs is one of the largest sections of the State Department. Its many responsibilities include issuing passports and assisting citizens overseas by putting out travel alerts and helping with emergency services. The bureau is also tasked with granting temporary visas to foreigners who want to visit or work in the United States.
"It would be a huge mistake," said Anne Richard, who led the bureau of Population Refugees, and Migration during President Barack Obama's second term.
The proposals were written in a memo submitted to the White House Office of Management and Budget from the White House Domestic Policy Council as part of President Trump's March executive order pushing for ideas for Government Reorganization.
A State Department spokesperson referred CNN to the White House for comment.
A senior White House official cautioned that the proposal was far from becoming policy, telling CNN that the idea of moving the longstanding State Department bureaus to the Department of Homeland Security is "one among many in a document resulting from a brainstorming session focused on improving efficiencies across government. None has been reviewed in great depth, let alone formally approved."
The official added that the memo "was a first step in exploring all options to make the government work better. No doubt many of the ideas will end up on the cutting-room floor, while others will ultimately may be adopted."
The White House did not explain the reasoning behind the recommendations, but perhaps the idea is rooted in a desire to streamline the refugee vetting process. As it stands currently, the United Nations High Commission of Refugees refers applicants to the State Department for vetting.
Nine international Resettlement Support Centers (RSC), all of which are managed by the Bureau of Population, Refugees and Migration, prepare applicants for the vetting process. Prospective refugees are then sent to the Department of Homeland Security for full vetting. After final vetting at DHS, the State Department then resumes supervision of the process with its Reception and Placement Program.
It is also not entirely clear whether Congress would have to sign off on such a move.
Richard, the former Obama State Department official, pinned the proposal on an "imperfect understanding" of the bureau's function. It's not mainly a law enforcement body, Richard said, rather, it works with nongovernmental organizations and the UN to assist refugees around the world.
While the refugee resettlement program does fall under the bureau, Richard noted that DHS plays a role in vetting potential refugees before they come to the United States. She said it would be a mistake for the government to neglect initiatives like the bureau she used to lead over what she called a "real but small" risk related to fighting terrorism.
"We could be a lot smarter than this and the administration could be focused on facts and evidence," Richard said. "But instead the administration is making decisions based on people's fears."
CLARIFICATION: This story has been updated to give further detail on the role the Resettlement Support Centers play in the refugee vetting process.