ATLANTA — Prosecutors in Georgia appear increasingly likely to open a criminal investigation of President Trump over his attempts to overturn the results of the state’s 2020 election, an inquiry into offenses that would be beyond his federal pardon power.

The new Fulton County district attorney, Fani Willis, is already weighing whether to proceed, and among the options she is considering is the hiring of a special assistant from outside to oversee the investigation, according to people familiar with her office’s deliberations.

At the same time, David Worley, the lone Democrat on Georgia’s five-member election board, said this week that he would ask the board to make a referral to the Fulton County district attorney by next month. Among the matters he will ask prosecutors to investigate is a phone call Mr. Trump made in which he pressured Georgia’s secretary of state to overturn the state’s election results.

Jeff DiSantis, a district attorney spokesman, said the office had not taken any action to hire outside counsel and declined to comment further on the case.

Some veteran Georgia prosecutors said they believed Mr. Trump had clearly violated state law.

“If you took the fact out that he is the president of the United States and look at the conduct of the call, it tracks the communication you might see in any drug case or organized crime case,” said Michael J. Moore, the former United States attorney for the Middle District of Georgia. “It’s full of threatening undertone and strong-arm tactics.”

He said he believed there had been “a clear attempt to influence the conduct of the secretary of state, and to commit election fraud, or to solicit the commission of election fraud.”

The White House declined to comment.

Mr. Worley said in an interview that if no investigation had been announced by Feb. 10, the day of the election board’s next scheduled meeting he would make a motion for the board to refer the matter of Mr. Trump’s phone calls to Ms. Willis’s office. Mr. Worley, a lawyer, believes that such a referral should, under Georgia law, automatically prompt an investigation.

If the board declines to make a referral, Mr. Worley said he would ask Ms. Willis’s office himself to start an inquiry.

Brad Raffensperger, the secretary of state, is one of the members of the board and has said that he might have a conflict of interest in the matter, as Mr. Trump called him to exert pressure. That could lead him to recuse himself from any decisions on a referral by the board.

Mr. Worley said he would introduce the motion based on an outside complaint filed with the state election board by John F. Banzhaf III, a George Washington University law professor.

Mr. Banzhaf and other legal experts say Mr. Trump’s calls may run afoul of at least three state criminal laws. One is criminal solicitation to commit election fraud, which can be either a felony or a misdemeanor.

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.”

“My feeling based on listening to the phone call is that they probably will see if they can get it past a grand jury,” said Joshua Morrison, a former senior assistant district attorney in Fulton County who once worked closely with Ms. Willis. “It seems clearly there was a crime committed.”

He noted that Fulton County, which encompasses much of Atlanta, is not friendly territory for Mr. Trump if he were to face a grand jury there.

The inquiry, if it comes to pass, would be the second known criminal investigation of Mr. Trump outside of federal pardon power. He is already facing a criminal fraud inquiry into his finances by the Manhattan district attorney, Cyrus R. Vance Jr. Even Georgia’s Republican governor, Brian Kemp, does not have the power to pardon at the state level, though it’s not assured that he would issue a pardon anyway, given his frayed relationship with Mr. Trump. Nonetheless, in Georgia, pardons are handled by a state board.

The question of whether or not to charge the nation’s 45th president would present a unique challenge for any district attorney. Ms. Willis, who took office only days ago, is a seasoned prosecutor not unaccustomed to the limelight and criticism. A graduate of Howard University and the Emory University School of Law in the Atlanta area, she is the first woman, and the second African-American, to hold the job of top prosecutor in Fulton County, Georgia’s most populous, with more than one million residents.

Ms. Willis, 49, is known for the leading role she played in the 2015 convictions of 11 educators in a standardized-test cheating scandal that rocked Atlanta’s public school system. She is taking office at a time when Atlanta, like other big cities, is seeing a rise in crime.

She must also deal with the high-profile fatal shooting of a Black man, Rayshard Brooks, by a white police officer in June 2020 and has said she will take a fresh look at charges brought against the officer by her predecessor.

Several calls by Mr. Trump to Georgia Republicans have raised alarms about election interference. In early December, he called Mr. Kemp to pressure 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 pressure campaign culminated in a Jan. 2 call by Mr. Trump to Mr. Raffensperger. “I just want to find 11,780 votes,” Mr. Trump said on the call, during which Mr. Raffensperger and his aides dismissed the president’s baseless claims of fraud.

After the Jan. 2 call, a complaint was sent to the election board by Mr. Banzhaf. (Three of his law students once brought a complaint that forced former Vice President Spiro Agnew to pay back to the state of Maryland money he had received as kickbacks.) Mr. Banzhaf has subsequently supplemented his complaint to incorporate the call made to the Georgia election investigator.

The complaint was also sent to Ms. Willis, and to Chris Carr, the Republican attorney general; a spokesperson for Mr. Carr could not be reached Friday.

Of the three Republicans on the board besides Mr. Raffensperger, one of them, Rebecca N. Sullivan, did not return a phone call, and another, Anh Le, declined to comment. The third, T. Matthew Mashburn, said that it would be inappropriate for him to comment on how he would vote before the motion was presented.

However, Mr. Mashburn also said that he was troubled by some of the language Mr. Trump had used in his phone call to Mr. Raffensperger. Mr. Mashburn noted, in particular, a moment when the president told Mr. Raffensperger, “There’s nothing wrong with saying that, you know, um, that you’ve recalculated.”

“The use of the word ‘recalculate’ is very dangerous ground to tread,” Mr. Mashburn said.