If confirmed, Mr Barr will replace Jeff Sessions, who was sacked from the role last month.
Mr Sessions recused himself from the investigation into alleged Russian interference in the 2016 presidential election, helping to set in motion the appointment of special counsel Robert Mueller, which in turn created deep tensions between Mr Trump and his justice department.
The president tweeted that Mr Barr was “one of the most highly respected lawyers and legal minds in the country”.
His potential appointment is already proving controversial, however.
Last year, Mr Barr wrote an article in the Washington Post defending the president’s decision to fire former FBI director James Comey.
That decision is being examined by Robert Mueller as a possible obstruction of justice.
Mr Barr also said there was more reason to investigate a uranium deal approved while Hillary Clinton was secretary of state than potential collusion between Russia and the Trump campaign.
Democratic leader in the House of Representatives, Nancy Pelosi, described the nomination as “deeply concerning”, adding that Mr Barr must allow Mr Mueller’s investigation to continue unimpeded.
In a separate announcement, Donald Trump nominated former Fox News presenter Heather Nauert to be the next US ambassador to the United Nations.
The president described Ms Nauert, who has been state department spokeswoman since April 2017, as “very talented, very smart, very quick”, adding: “I think she’s going to be respected by all.”
If confirmed by the Senate, she will take over from Nikki Haley, who announced in October that she would be stepping down at the end of the year.
Before serving under two secretaries of state – Rex Tillerson and Mike Pompeo – Ms Nauert had little foreign policy experience.
She has been the frontrunner for the UN post for some weeks and last month President Trump said she was “under very serious consideration”.
He added: “She’s excellent, she’s been with us a long time, she’s been a supporter for a long time, really – excellent.”
Ms Haley has held the role since November 2016.
When she announced her decision to step down, the president called her a “very special person” and said she had told him some months before that she wanted to take some time off.
Ms Haley has made no secret of the fact that she has had policy disagreements with Mr Trump.
In a Washington Post article in September, she wrote: “I proudly serve in this administration, and I enthusiastically support most of its decisions and the direction it is taking the country.
“But I don’t agree with the president on everything.”