WASHINGTON—President Donald Trump’s shake-up of his national-security team is likely to reshape the role played by Jared Kushner, who has been a key counselor but whose global portfolio created friction with departing Secretary of State Rex Tillerson, administration officials and others close to him said.

But the arrival of Mike Pompeo as secretary of state and John Bolton as national security adviser could curb Mr. Kushner’s sway if foreign diplomats, who have been courting the president’s son-in-law as an alternative to...