ATLANTA — The office of Georgia’s secretary of state, Brad Raffensperger, on Monday started an investigation into former President Donald J. Trump’s attempts to overturn the state’s election results, including a phone call he made to Mr. Raffensperger in which Mr. Trump pressured him to “find” enough votes to reverse his loss.

Such inquiries are “fact-finding and administrative in nature,” the secretary’s office said, and are a routine step when complaints are received about electoral matters. Findings are typically brought before the Republican-controlled state board of elections, which decides whether to refer them for prosecution to the state attorney general or another agency.

The move comes as Fani Willis, the Democratic district attorney of Fulton County, which encompasses much of Atlanta, is weighing whether to begin a criminal inquiry of her own. A spokesman for Ms. Willis declined to comment on Monday.

The January call was one of several attempts Mr. Trump made to try to persuade top Republican officials in the state to uncover instances of voting fraud that might change the outcome, despite the insistence of voting officials that there was no widespread fraud to be found. He also called Gov. Brian Kemp in early December and pressured him to call a special legislative session to overturn his election loss. Later that month, Mr. Trump called a state investigator and pressed the official to “find the fraud,” according to those with knowledge of the call.

“The Secretary of State’s office investigates complaints it receives,” Walter Jones, a spokesman for the office, said in a statement on Monday. “The investigations are fact-finding and administrative in nature. Any further legal efforts will be left to the Attorney General.”

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.

Former prosecutors said Mr. Trump’s calls might run afoul of at least three state laws. One is criminal solicitation to commit election fraud, which can be either a felony or a misdemeanor; as a felony, it is punishable by at least a year in prison. There is also a related conspiracy charge, which can be prosecuted either as a misdemeanor or a felony. A third law, a misdemeanor offense, bars “intentional interference” with another person’s “performance of election duties.”

Jason Miller, a senior adviser to Mr. Trump, said in a statement: “There was nothing improper or untoward about a scheduled call between President Trump, Secretary Raffensperger and lawyers on both sides. If Mr. Raffensperger didn’t want to receive calls about the election, he shouldn’t have run for secretary of state.”

Mr. Biden’s victory in Georgia was reaffirmed after election officials recertified the state’s presidential election results in three separate counts of the ballots: the initial election tally; a hand recount ordered by the state; and another recount, which was requested by Mr. Trump’s campaign and completed by machines. The results of the machine recount show Mr. Biden won with a lead of about 12,000 votes.

Mr. Biden was the first Democrat to win the presidential election in Georgia since 1992. Mr. Trump accused Mr. Kemp and Mr. Raffensperger, both Republicans, of not doing enough to help him overturn the result in the weeks after the election. Mr. Kemp and Mr. Raffensperger had each resisted numerous attacks from Mr. Trump, who called the governor “hapless” and called on the secretary of state to resign.

Maggie Haberman contributed reporting.