“It just makes it neat and clean at the beginning of the quarter,” said one of Mr. Paul’s associates. Aides to Mr. Paul said Tuesday that they had no comment.

The senator’s associates said he would most likely declare his candidacy in Kentucky, the state that elected him in 2010 when he was still an ophthalmologist with no experience in government. He would then depart on a campaign swing through states with the earliest nominating contests, like Iowa, New Hampshire and South Carolina.

One issue Mr. Paul has yet to resolve is how he can run for both president and senator at the same time. State law in Kentucky bars candidates from seeking two offices at once.