WASHINGTON—Thousands of unaccompanied minors who crossed the border illegally aren’t being tracked after being placed with sponsors by the Health and Human Services Department, according to a bipartisan Senate investigation that faulted the HHS and Homeland Security departments for the lapse.

Representatives from HHS and the Justice Department pushed back in a Senate subcommittee hearing Thursday, with an HHS representative saying the agency has started doing background checks on adults that volunteered to take care of the...