Yet what worries some of Mr. Paul’s friends is that for all his early moves to cultivate donors he is not doing enough to build the political network necessary to mount a viable presidential campaign, and that he is relying more on instinct and improvisation than considered strategy.

Last fall, for example, when Mr. Paul attended a Republican National Committee fund-raiser at the Central Park West penthouse of the New York Jets owner Woody Johnson, the senator launched into his standard speech about how Republicans could appeal more to younger voters by abandoning a no-tolerance approach to drug crimes. Some in attendance were taken aback, given that Mr. Johnson’s brother died of a drug overdose and his daughter, who battled drugs and alcohol, died at age 30 in 2010. Mr. Paul’s staff had not briefed him on the background.

His efforts to build a political organization also have gotten off to a halting start. Mr. Paul’s aides interviewed a well-regarded Republican operative named Dorinda Moss, who had served as the finance director of the National Republican Senatorial Committee. But she eventually went to work for Mr. Rubio, in part because Mr. Paul never got around to offering her a job.