The Ajax boss is set to leave his role with the Dutch champions at the end of the Eredivisie season, according to The Athletic.
A verbal agreement has been reached between United and Ten Hag, the report says, but a four-year contract is likely to follow.
United still need to finalise documentation and further details, and Ajax will not confirm any agreement or announcement until after they PSV Eindhoven on April 17 in the Dutch Cup final.
Ten Hag's coaching staff will be confirmed in due course, but Chris Armas, Ewan Sharp and Sascha Lense who were all brought in by Rangnick, will leave at the end of the season.
Mike Phelan, who returned to the club last year under Ole Gunnar Solskjaer, having previously been assistant manager to Sir Alex Ferguson, is also likely to depart.
Manchester United and Ten Hag's representatives SEG have declined to comment so far.
Ralf Rangnick has been in interim charge of the Red Devils since November following the departure Solskjaer.
Ten Hag will takeover from the German, with United opting against appointing the former Tottenham Hotspur and now Paris Saint-Germain manager Mauricio Pochettino.
Dutchman Ten Hag joined Ajax in December 2017 from Bayern Munich, having previously worked under the stewardship of Manchester City boss Pep Guardiola.
The 52-year-old helped the Dutch champions to reach the semi-finals of the Champions League in 2019, and has won the Eredivisie title in 2019 and 2021. His side are currently four points ahead at the top of the Eredivisie table.
Manchester United are currently seventh in the Premier League following their 1-0 defeat to Everton on Saturday. They remain six points off the Champions League places.