David Worley, the sole Democrat on the state elections board, said Monday that administrative inquiries by the secretary of state’s office could result in criminal charges.

“Any investigation of a statutory violation is a potential criminal investigation depending on the statute involved,” he said, adding that in the case of Mr. Trump, “The complaint that was received involved a criminal violation.”

Mr. Worley said that now that an inquiry had been started by the secretary of state’s office, he would not introduce a motion at Wednesday’s state board of election meeting, as he had originally planned to do, in an effort to refer the case to the Fulton County district attorney’s office.

Not long after the call to Mr. Raffensperger became public, several complaints were filed. One came from John F. Banzhaf III, a George Washington University law professor.