Mr. Trump’s aides were on hand for the meeting, and the president told Mr. Manchin that a background checks bill that the senator had pushed for with a Republican counterpart, Senator Patrick J. Toomey, Republican of Pennsylvania, was still on the table, according to the people briefed on the discussion.

But the polling data, White House aides said, indicated that the issue does not help the president with his core base of supporters, according to the people briefed on the meeting.

People familiar with the meeting said a number of different options of gun measures were discussed. But those familiar with the meeting said the president’s likeliest course of action was a menu of smaller items, like a slimmer version of a background checked bill and “red flag” laws, which allow the authorities to temporarily confiscate firearms from those who are found by a judge to be a danger to themselves or to others.