Trump offered Kudlow the job Tuesday night over the phone and he accepted, the source said.
White House press secretary Sarah Sanders later confirmed Kudlow's hire in a statement to reporters.
"Larry Kudlow was offered, and accepted, the position of assistant to the President for Economic Policy and Director of the National Economic Council," Sanders said. "We will work to have an orderly transition and will keep everyone posted on the timing of him officially assuming the role."
The move makes Kudlow Trump's second top economic adviser after Gary Cohn, a former Goldman Sachs executive who worked for Trump for over a year, announced his resignation earlier this month over internal disagreements around the President's decision to impose tariffs on steel and aluminum imports.
Trump acknowledged his disagreement on tariffs with Kudlow in a conversation with reporters on Tuesday, but said he welcomed the difference of opinion.
"I'm looking at Larry Kudlow very strongly. I've known him a long time. We don't agree on everything but in this case I think that's good. I want to have a divergent opinion -- we agree on most," Trump said.
He added that Kudlow has "come around to believing in tariffs as a negotiating point."
Peter Navarro, Trump's top trade adviser and a fervent proponent of the President's tariffs, was seen as the force behind Cohn's eventual departure. The two butted heads over the decision and internally Trump's decision was seen as a win for Navarro and a damaging loss for Cohn.
Wary of the narrative that Navarro and Kudlow will clash in the same way that Navarro and Cohn did, a source close to Navarro said the President's trade adviser is "very happy" with Kudlow's selection.
The source said the two men have been friends for years, dating back to the time they spent on CNBC sets together as economic analysts.
While the two men differ on trade, the source said Navarro and Kudlow agree on nearly every other economic issue.
But Navarro publicly criticized Kudlow's views on trade in an interview with CNN earlier this month.
"Larry and Steve have never, ever supported the President on trade," Navarro told CNN's Jake Tapper. Although the trade adviser said that the three were "brothers in arms during the campaign" they are "dead wrong on the economics" of trade.
Another source said Kudlow and Treasury Secretary Steven Mnuchin have a much friendlier relationship than Mnuchin had with Cohn. They worked together on Trump's economic plan during the presidential campaign, the source added.
Trump hinted that Kudlow was his pick at the time, telling reporters that the media personality "has a good chance" at getting the job.
Trump spoke to Kudlow multiple times over the past week, a separate person familiar with the matter said, but the economist was not officially offered the job until Tuesday.
The selection quickly received plaudits from Capitol Hill Republicans who have long known Kudlow as a media analyst and regular at Republican conferences.
"Couldn't be more pleased," said Tom Cole, an Oklahoma Republican. "He is somebody that House Republicans are extraordinarily comfortable with."
CNN's Jeremy Diamond and Kaitlan Collins contributed to this report.