Darlow made 22 Premier League appearances for the West Yorkshire club last season, finishing the campaign as Daniel Farke's first-choice goalkeeper, and is also Wales' current number one.
The move is likely to trigger the exit of Altay Bayindir, who spent much of last season as understudy to Manchester United's first-choice Senne Lammens.
With Radek Vitek also expected to leave, either on loan or permanently after an impressive spell at Bristol City, manager Michael Carrick needed to add further goalkeeping cover, given 40-year-old Tom Heaton has not played a first-team game since February 2023.
Providing the deal goes through as expected, Darlow will be involved in United's pre-season campaign, which begins against Wrexham in Helsinki on 18 July.
The move is a notable one given the rivalry between the two clubs, with Darlow set to become one of only a handful of players to switch directly from Leeds to United in the modern era. According to The Athletic, there are still finer details to sort out, but the two parties are in advanced stages of finalising terms.
United had also considered Wolves' Sam Johnstone, who started his career in the club's academy, but appear to have settled on Darlow as their preferred option given his Premier League experience and homegrown status.
Darlow replaced Lucas Perri as Leeds United's first-choice goalkeeper at the start of the year and played a key role in helping the club maintain their Premier League status last season. Everton had also explored a move for the 15-times capped Wales international before United moved to secure his signature as competition and cover for Lammens heading into a campaign that will include Champions League football.
United's goalkeeping department has undergone significant change over the past year, with the club also agreeing to send Andre Onana back to Trabzonspor for a second season-long loan in the Turkish top flight, continuing a reshuffle that has seen the position rebuilt almost entirely around Lammens.