Mr. Graham would enter the race a long-shot candidate, but his expected focus on national security could alter the discussion around those issues, forcing a crop of GOP governors and less-experienced lawmakers to sharpen their foreign-policy positions.

“No one really begins to have the kind of foreign-policy experience in the current lineup of candidates that Lindsey does,” said Sen. Dan Coats (R., Ind.). “They’re going to have to up their game if they want to go toe-to-toe with Lindsey.”

Mr. Graham’s entrance into the race also could shake up other candidates’ calculations on how heavily to invest in the primary in the senator’s home state of South Carolina, which holds one of the earliest elections. Mr. Graham narrowly led the field of GOP candidates in an NBC News/Marist poll of Republican-leaning voters in South Carolina earlier this month, but he landed at the end of the pack in Iowa.

“I have questions about how successful Lindsey Graham could be in Iowa, but I do think he could be very successful in pushing foreign-policy issues and steering the foreign-policy debate,” said Craig Robinson, former political director of the Iowa Republican Party.