Hillary Clinton’s campaign isn’t considering primary rival Bernie Sanders as her running mate, but is actively looking at Sen. Elizabeth Warren, whose populist politics line up closely with Mr. Sanders, people familiar with the process said.

The vetting remains in its early stages. So far, potential candidates have been scrutinized using publicly available information. The Clinton team hasn’t asked anyone to submit tax returns or other personal information, one of the people said. Conversations with Mrs. Clinton herself about...