RALEIGH, N.C. — Donald J. Trump’s already short list of potential running mates got shorter on Wednesday when Senator Bob Corker removed himself from contention a day after appearing at an audition of sorts with the presumptive Republican nominee.

Mr. Corker, Republican of Tennessee, told The Washington Post in an interview that he had told Mr. Trump during their travels together that he did not want to be considered for the vice-presidential nomination.

Another contender whom Mr. Trump’s aides were said to have preliminarily vetted, Senator Joni Ernst of Iowa, also said on Wednesday that she was not looking to be Mr. Trump’s running mate. But a person close to the Trump campaign who asked for anonymity to discuss the vetting process said the campaign had indicated to Ms. Ernst and Mr. Corker that they would not end up on the final list. An aide to Mr. Corker disputed that account.

The developments came amid a highly unusual selection process for a running mate: testing out candidates in the court of public opinion, an approach that feels oddly similar to the one Mr. Trump took on his reality television show, “The Apprentice.” Moving away from his initial hope to unveil the running mate at the Republican National Convention in Cleveland, Mr. Trump is instead maximizing the drama — and public attention — around his vice-presidential selection process.