The White House contacted Ms Barrett on September 19, the day after Justice Ruth Bader Ginsburg died creating the court vacancy, according to formal paperwork submitted to the Senate.
Mr Trump made the offer when she visited the White House on Monday September 21, five days before the president made a formal announcement of his preference.
The details were revealed in a report submitted to the Senate Judiciary Committee submitted by the White House.
It comes as Ms Barrett held a day of private meetings at the Capitol, drawing praise from Republican senators but opposition from Democrats objecting to her conservative views and fast-track confirmation before the election on November 3.
Senate Majority Leader Mitch McConnell said he was “even more convinced” of Ms Barrett after their brief meeting.
Noting she is a working mother of seven children, he scoffed at Democratic objections that the judge would put Americans’ access to health care at risk or turn back the clock on women’s rights.
“What a joke,” he said.
As questions swirl about the potential for the results of the presidential election between Mr Trump and Democrat Joe Biden to wind up in a court battle, Ms Barrett offered no suggestion she would recuse herself from hearing any such election cases.
Leading Democrats say Ms Barrett should not hear any cases about the presidential election because her nomination is the first in US to come so close to one, with early voting underway.
But Republicans said Ms Barrett, if confirmed, should absolutely be involved.
Senator Ted Cruz said that is “the entire reason” why the Senate should rush to fill the vacant seat — “so that the Supreme Court can resolve any cases that arise in the wake of the election”.
But Democrats are confronting the limits of their power as they fight against the nomination.
With Republicans holding a 53-47 Senate majority, and just two Republican senators opposing a quick vote, Ms Barrett appears to have enough support for confirmation.
Senate Democratic leader Chuck Schumer acknowledged on Tuesday it will be an “uphill fight” to stop Mr Trump’s nominee.
But he said Americans are on Democrats’ side in preferring to wait until after the election so the winner can choose the next justice.