Stavridis declined to comment through a spokesman and directed inquiries to the Clinton campaign. The Clinton campaign also declined to comment.
Donald Trump, the presumptive Republican nominee, is reportedly also considering a former military officer, retired Lt. Gen. Michael T. Flynn, in his vice presidential search.
Stavridis adds a lesser-known figure to a list of mostly politicos — current and former lawmakers or government officials — who are known to be considered by Clinton.
Stavridis, 61, served as the supreme allied commander of NATO and commander of U.S. European Command from 2009 to 2013. And he chaired the U.S. Naval Institute -- an independent group that serves as an outside source of ideas for the U.S. Navy -- after retiring.
Just before Stavridis retired from the Navy, he was under internal investigation for his use of a military plane to fly with his wife and some members of his staff to France to attend an event hosted by an international society of Burgundy wine enthusiasts in 2010. The inspector general also investigated whether he and others billed the government for non-official expenses. Stavridis was cleared of wrongdoing.
Ray Mabus, secretary of the Navy, concluded after the investigation that Stavridis "never attempted to use his public office for private gain nor did he commit personal misconduct."
Stavridis is the author of several books, including a memoir: "The Accidental Admiral: A Sailor Takes Command at NATO."
He served as military assistant to Richard Danzig, who was Navy secretary under President Bill Clinton.
"I regard him as one of the most talented and admirable people I know," Danzig said of Stavridis in a statement to The Washington Post. "Admiral Stavridis is a man of remarkable integrity, character and range of talent.
"Alongside impressive achievements as a warrior and diplomat in Asia, South America and Europe, he has written reflectively and shown his patriotism and generosity by counseling — and inspiring — generations of naval officers," added Danzig, who also was an adviser to Barack Obama's presidential campaign.
It's not clear whether Stavridis has ever expressed interest in politics.
According to Anne-Marie Slaughter, who encountered Stavridis when she worked under Clinton at the State Department, he is considered to be something of a "soldier scholar."
"He's also a thinker and a writer ... he combines strategic ability with leadership ability," Slaughter said in an interview. "He's very highly regarded by all the people I know who know him."