Donald Trump is tapping a veteran of George W. Bush’s White House to serve as his main adviser on terror threats and cyberattacks, while taking a step aimed at ensuring the new appointee has the internal clout needed to succeed in the role.

The Trump transition team announced Tuesday that Thomas P. Bossert will serve as an adviser to Mr. Trump on national security, terrorism and cyber issues, putting him on the same level as the powerful incoming national security adviser, Michael Flynn.

“It’s...