The State Department is continuing an investigation of email use among employees who worked for Hillary Clinton, the former secretary of state, asking scores of current and former officials to submit to questioning by the bureau overseeing diplomatic security, former officials said on Sunday.

The investigation is examining whether the employees used secure channels and the proper classification designations for what appeared to be routine emails at the time, the former officials said. The emails were on subjects that were not considered classified at the time, but that have been or are being retroactively marked as classified.

The inquiry potentially pushes back into the spotlight a deeply political issue that President Trump used as a weapon in his 2016 presidential campaign, and has repeatedly returned to during his time in office, despite the F.B.I. closing a previous investigation without finding wrongdoing by Mrs. Clinton. Mr. Trump’s insistence on raising the issue reflects a pattern in which the president pushes his own narrative even if an investigation or an analysis by an agency in his own administration has concluded otherwise.

The emails were sent to Mrs. Clinton while she was secretary of state from 2009 to 2013, in President Barack Obama’s administration. They appear to have come to the attention of the diplomatic security bureau during the earlier inquiries conducted by the State Department, Congress and the F.B.I. into Mrs. Clinton’s use of a private email server.