President Donald Trump will push ahead with legal challenges to the results of last week’s election after US attorney-general William Barr told federal prosecutors to look into any “substantial” allegations of voting irregularities.

Barr’s directive to prosecutors prompted the top lawyer overseeing voter fraud investigations to resign in protest. It came after days of attacks on the integrity of the election by Trump and Republican allies, who have alleged widespread voter fraud, without providing evidence.

Trump has not conceded the election to Democrat Joe Biden, who on Saturday secured the more than the 270 votes in the Electoral College needed to win the presidency.

The Trump campaign has filed several lawsuits claiming the election results were flawed. Judges have tossed out lawsuits in Michigan and Georgia, and experts say Trump’s legal efforts have little chance of changing the election result.

Barr told prosecutors on Monday that “fanciful or far-fetched claims” should not be a basis for investigation and his letter did not indicate the justice department had uncovered voting irregularities affecting the outcome of the election.

But he did say he was authorising prosecutors to “pursue substantial allegations” of irregularities of voting and the counting of ballots.

Richard Pilger, who for years has served as director of the Election Crimes Branch, announced in an internal email he was resigning from his post after he read “the new policy and its ramifications”.

The previous justice department policy, designed to avoid interjecting the federal government into election campaigns, had discouraged overt investigations “until the election in question has been concluded, its results certified, and all recounts and election contests concluded”.

Biden’s campaign said Barr was fuelling Trump’s far-fetched allegations of fraud. “Those are the very kind of claims that the President and his lawyers are making unsuccessfully every day.”