Mr. Trump has declined to consider a group of up-and-coming Hispanic Republicans, including Govs. Brian Sandoval of Nevada and Susana Martinez of New Mexico, as well as Senator Marco Rubio of Florida. People close to all three, on the condition of anonymity, confirmed that they are not being vetted for vice president.

He has skipped over leaders who might have delivered a crucial state in the Electoral College: Ohio. Gov. John Kasich and Senator Rob Portman, Ohio politicians with deep experience in Washington, are not under consideration, according to aides. Nor is Gov. Scott Walker of Wisconsin, who became a hero to conservatives for his clashes with organized labor.

Mr. Trump has also overlooked prominent Republican women, including Senator Kelly Ayotte of New Hampshire, a foreign policy hawk who is well regarded by national party leaders. Neither she nor Ms. Haley is being vetted.

Rob Godfrey, a spokesman for Ms. Haley, said the governor declared in May that she did not want the job and has not budged. “Since the governor made it clear that she was not interested in serving as anyone’s vice president, there was no reason for her to participate in the vetting process,” he said.