“I just want to find 11,780 votes,” Mr Trump told Republican Secretary of State Brad Raffensperger in a recording released by the Washington Post.
Mr Raffensperger is heard replying that Georgia’s results were correct.
Joe Biden won Georgia alongside other swing states, winning 306 electoral college votes to Mr Trump’s 232.
Since the 3 November vote, Mr Trump has been alleging widespread electoral fraud without providing any evidence.
All 50 states have certified the election result, some after recounts and legal appeals. So far, US courts have rejected 60 challenges to Mr Biden’s win.
Congress is due to formally approve the election result on 6 January.
Mr Biden, a Democrat, is due to be inaugurated as president on 20 January.
Voters in Georgia are due to vote again on Tuesday to elect two senators for the state. The result could determine the balance of power in the Senate.
If the two Democrat contenders win, then there will be equal numbers of Republican and Democratic senators, and Kamala Harris, the Democratic vice-president-elect, will have the deciding vote.
Mr Biden’s Democrats already control the lower House of Representatives.
In other developments, all 10 living former US defence secretaries have penned an opinion piece in the Washington Post, urging President Trump not to question the election results.
The group also said he should not involve the military in voting disputes, an idea that’s been mooted by some of Trump’s supporters.
What happened during the Trump call?
In excerpts released by the Washington Post, Mr Trump can be heard alternately cajoling and pressurising Georgia’s secretary of state.
He insisted that he had won the election in Georgia and told Mr Raffensperger that there was “nothing wrong with saying you have recalculated”.
Mr Raffensperger responded by saying: “The challenge you have Mr President is that the data you have is wrong.”
Later in the call Mr Trump said the rumour was that ballots had been shredded and voting machinery had been removed from Fulton County in the state – a charge Mr Raffensperger’s lawyer said was not the case.
The president then threatened the official with possible legal consequences.
“You know what they did and you’re not reporting it. That’s a criminal offence. You can’t let that happen. That’s a big risk to you and to Ryan, your lawyer,” Mr Trump said.
He then called for the extra 11,780 votes – which would have given him a total of 2,473,634 votes in the state, one more than Mr Biden, who received 2,473,633 votes.
He told Mr Raffensperger he should re-examine the result in the state.
“You can re-examine it, but re-examine it with people who want to find answers, not people who don’t want to find answers,” he said.
“Mr President, you have people who submit information and we have our people that submit information and then it comes before the court and the court has to make a determination,” Mr Raffensperger replied. “We have to stand by our numbers, we believe our numbers are right.”
On Sunday Mr Trump tweeted that Mr Raffensperger had not given details of the fraud the president alleges. “He has no clue!” the president tweeted.
Mr Raffensperger tweeted back: “Respectfully, President Trump: What you’re saying is not true. The truth will come out.”
Respectfully, President Trump: What you're saying is not true. The truth will come out https://t.co/ViYjTSeRcC
— GA Secretary of State Brad Raffensperger (@GaSecofState) January 3, 2021
The White House has not commented on the release of the audio.
Speaking in Georgia, Kamala Harris called Trump’s comments “a bold abuse of power.”
Moderate Republican Adam Kinzinger also tweeted: “This is absolutely appalling. To every member of Congress considering objecting to the election results, you cannot – in light of this – do so with a clean conscience.”
When will the election result be certified?
There is a developing split in the Republican party after 12 senators said they would not vote to certify Mr Biden’s election victory in a Senate session on 6 January.
Ted Cruz led a group of 11 senators calling for a 10-day delay to audit unsubstantiated allegations of election fraud.
Vice-President Mike Pence – who as president of the Senate is due to oversee the session and declare Mr Biden the winner – said he welcomed the move.
He stopped short of repeating allegations of fraud – but his chief of staff said Mr Pence shared what he called “the concerns of millions of Americans about voter fraud and irregularities”.
Missouri Senator Josh Hawley has also said he will reject the result over concerns about the integrity of the election.
Meanwhile four senators including Mitt Romney have signed a statement saying they will vote to certify Mr Biden’s win.
Top Republicans have said the Senate’s role in certifying the election is largely ceremonial and should not be an opportunity for further lengthy debate about the result.
Senate Majority Leader Mitch McConnell has already recognised Mr Biden’s victory and has asked other Republicans not to object.
A group of Republicans in the lower chamber of Congress, the House of Representatives, is also planning to contest the election results.
On Sunday a new Congress convened for the first time, with senators and representatives being sworn in. Democrat Nancy Pelosi was narrowly re-elected as House Speaker.
The House voted 216-209 for Ms Pelosi. Democrats lost 11 seats in the November elections so their majority is 222-212. Five Democrats chose not to support Ms Pelosi – two voted for lawmakers who were not running and three more voted “present”.
What will happen on 6 January?
Objections that are endorsed by a member of the House of Representatives and a member of the Senate must be considered by lawmakers in a two-hour debate, followed by a vote.
However, for an objection to be upheld, a majority in both chambers must vote in favour. Republicans hold the majority in the Senate but some of their number have already said they will not contest the results.
Democrats are in the majority in the House.