The near miss is described in a new book co-written by the agent, Gerald Blaine, who was stationed outside the new president's house in Washington the evening after the assassination in November 1963.
In the early hours of the next morning, Mr Blaine heard someone approaching and drew his Thompson submachine gun, the book, The Kennedy Detail, explains.
It adds: "He firmly pushed the stock into his shoulder, ready to fire.
"He'd expected the footsteps to retreat with the loud sound of the gun activating, but they kept coming closer. Blaine's heart pounded, his finger firmly on the trigger. Let me see your face, you -------.
"The next instant, there was a face to go with the footsteps. The new President of the United States, Lyndon Baines Johnson, had just rounded the corner, and Blaine had the gun pointed directly at the man's chest. In the blackness of the night, Mr Johnson's face went completely white.
"A split second later, Blaine would have pulled the trigger," it goes on.