The official, Richard Pilger, would have overseen such investigations, according to BBC reports.
Any such cases would normally be the remit of individual states, but Mr. Barr said this was not a hard and fast rule.
Donald Trump refuses to accept Joe Biden’s projected victory, and has made unsubstantiated fraud claims.
The president’s campaign is seeking an emergency injunction in Pennsylvania to prevent Mr Biden’s victory being certified in the state.
The president-elect’s projected win there on Saturday took him over the threshold of 270 electoral college votes needed to secure victory nationwide.
Mr. Trump’s spokeswoman vowed the legal battle to contest Mr Biden’s victory was only just beginning.
“This election is not over,” White House press secretary Kayleigh McEnany told a news conference. “Far from it.”
The attorney general wrote that inquiries could be made by federal prosecutors “if there are clear and apparently-credible allegations of irregularities that, if true, could potentially impact the outcome of a federal election in an individual State”.
Mr. Barr said prosecutors should only look into “substantial allegations” of irregularities, and that “specious, speculative, fanciful or far-fetched claims” should be ignored.
He acknowledged that individual states had the primary responsibility for the conduct of elections but said the justice department had “an obligation to ensure that federal elections are conducted in such a way that the American people can have full confidence in their electoral process and their government”.
The department would normally only go beyond preliminary investigations after an election had been concluded and the results certified, but Mr Barr said this could result in situations where “misconduct cannot realistically be rectified”.
Mr Pilger said he had quit in response to Mr Barr’s memo.
“Having familiarised myself with the new policy and its ramifications… I must regretfully resign from my role,” he wrote in an email to colleagues.
Separately, the Trump campaign filed a lawsuit on Monday in a Pennsylvania federal court, seeking an emergency injunction to stop state officials from certifying Mr. Biden’s victory in the state. The state’s Attorney General, Josh Shapiro, called the lawsuit “meritless”.
Prosecutors in Republican-controlled states meanwhile threw their weight behind the president’s challenge to the election results.
The 10 state attorneys general filed a so-called amicus brief at the US Supreme Court backing the Trump campaign’s case in Pennsylvania.
