Some close to Mrs. Clinton, the former secretary of state, say she was always likely to have someone with military experience on her vice-presidential short list, and Mr. Stavridis, currently the dean of the Fletcher School at Tufts University, fits the description.

During his four years as NATO’s supreme allied commander, he oversaw operations in the Middle East — Afghanistan, Libya, and Syria — as well as in the Balkans and piracy off the coast of Africa.

The Clinton campaign declined requests for comment, and Mr. Stavridis declined to comment other than referring calls to the campaign. The person with knowledge of the vetting spoke anonymously because of the sensitive nature of the process.