The Otley-born player, 23, who was freed by his boyhood club last month, is understood to be close to joining the Millers on a 12-month deal, subject to the successful completion of a medical.
Several rival Championship clubs had enquired about the services of White, who made 111 appearances during his time at Elland Road, where he worked under seven different managers or head coaches in eight years.
A number of League One clubs including Doncaster Rovers have also thrown their hat into the ring to sign White.
But the Millers, whose ex-Leeds contingent includes Paul Green, Frazer Richardson and former junior Danny Ward, appear to have won the race to sign him.
Ironically, White made his final appearance for Leeds on the last day of the 2014-15 season as a late substitute in the 0-0 home draw with the Millers on May 2.
White, who joined Leeds at the age of nine and made his debut at the age of 16 in August 2008, had been sidelined for the vast majority of the campaign with a metatarsal injury and was not offered an extension to his contract.
Despite showing him the exit door, Leeds were understood to have considered offering him fresh terms on a reduced wage, with White having secured a three-year deal worth around £9,000 a week in 2012.
Meanwhile, Leeds are understood to have enquired about the availability of Leicester City defender Liam Moore, currently on England U21 business and part of Gareth Southgate’s squad for the forthcoming UEFA European Championships in the Czech Republic.
Moore is down the pecking order at Leicester, who are set to allow him to leave the club on loan next season, with a number of Championship clubs interested in the 22-year-old, including Leeds.
United head coach Uwe Rosler signed the right-sided centre-half during his time in charge of Brentford in 2012-13.
Leeds have widened their search for new defensive recruits as they wait to see if former loan centre-half Sol Bamba will return to the club.
Leeds have an agreement in place with Bamba’s parent club Palermo and their interest in him does not been appear to have been affected by his scathing assessment of Massimo Cellino’s management of the club early last month.
United negotiated an option to take Bamba full-time when they signed him on a half-season loan in the January transfer window, with the Ivory Coast international impressing during the club’s upturn in the second half of the campaign.
The vastly-experienced 30-year-old defender has two years left on his Palermo contract, but he has made only one league appearance for the Italian club and is unlikely to be part of their future plans.