Donald Trump Jr., the president’s son who often speaks with his father about his views on gun laws, has raised concerns about both red-flag legislation and about tightening background checks, according to people familiar with the matter.

“Don Jr. is my gun expert,” Mr. Trump told supporters at a fundraiser Friday in Bridgehampton, N.Y., according to one of the people familiar with the matter. “He knows more about guns than anyone I know.”

Campaign advisers asked White House officials for at least a week to gauge how various options—including background checks and taking no action—would fare among the president’s base of supporters before the administration decides on an official course of action, a person close to the White House said.

The White House has also heard from prominent conservatives around the country who have urged the president not to take any major action on guns.