Sen. Lindsey O. Graham holds a news conference in Des Moines, Iowa, in February while exploring a presidential bid.  (Andrea Melendez/Des Moines Register via Associated Press)

This story has been updated.

Sen. Lindsey O. Graham (R-S.C.), who in recent months has moved closer to running for president, said Monday that he will announce if he will make a White House bid on June 1 from South Carolina.

"I will make an announcement on June 1st. You're all invited to come -- spend money when you do -- and I will tell you what I’m going to do about running for president," Graham said on CBS "This Morning."

Sen. Lindsey Graham (R-S.C.) says he will announce if he will make a White House bid on June 1 in South Carolina. (Reuters)

Although Graham initially declined to say if he is running when asked by the CBS hosts, he nonetheless indicated he is pursuing the GOP nomination – potentially declaring his candidacy ahead of schedule.

"I'm running because of what you see on television. I'm running because I think the world is falling apart. I've been more right than wrong on foreign policy," Graham said. "It's not the fault of others, their lack of this or that that makes me want to run. It’s my ability in my own mind to be a good commander in chief and to make Washington work. “

Graham formed an exploratory committee, Security Through Strength, in January and has been staffing it as he prepares for what looks increasingly like it will become a full-fledged run for president. Graham told South Carolina state lawmakers last week that he's "99.9 percent sure" he's going to run.

Politico reported earlier this month that Graham told donors that June 1 was the likely date for his presidential announcement. A Graham spokesperson did not immediately respond Sunday night to a question about his plans.

If Graham runs, he is likely to center his campaign around the theme of national security. He is among the most hawkish figures in the Republican Party and already has secured the support of Sen. John McCain (R-Ariz.), a close friend and fellow hawk.