The deal will see Sancho move to Stamford Bridge on a season-long loan with an obligation to buy for around £23m next summer.
Sancho is understood to have been eager to leave Old Trafford amid a fraught relationship with Erik ten Hag.
Mail Sport previously reported that both clubs were discussing a surprise swap deal involving Sancho and his former England teammate, Raheem Sterling, who was unexpectedly dropped from the Chelsea squad ahead of the Premier League start.
However, Sterling now appears poised to join Arsenal before Friday's transfer deadline.
Chelsea have reportedly reached an agreement over the signing of Jadon Sancho from Man United
After impressive performances during his first spell at Borussia Dortmund saw him break into the England team, United paid £73m to bring Sancho to Old Trafford in the summer of 2021.
However, under both Ole Gunnar Solskjaer and Ten Hag, the winger has failed to consistently replicate the performances that made him so vaunted before his move to the Premier League.
After a public falling out with the Dutchman saw him frozen out at United last season, Sancho returned to Dortmund on loan for the remainder of the 2023-24 campaign.
Back in the Bundesliga the winger once again demonstrated his potential, making 21 appearances in all competitions as Edin Terzic's side reached the final of the Champions League.