The 52-year-old is reportedly set to sign a four-year deal although a contract is yet to be signed.
Official confirmation of Ten Hag's appointment is not expected until after Ajax's Dutch Cup final against PSV Eindhoven this Sunday, according to British media reports.
Neither Manchester United, Ajax or Ten Hag's representatives, SEG, were available for comment when contacted by CNN.
Paris Saint-Germain's former Tottenham boss Mauricio Pochettino was another name linked with the job.
Ten Hag is touted as one of the most exciting managers in European football after building a reputation of success during his five years at Ajax.
Since his appointment during the 2017-18 campaign, he has led Ajax to Dutch league titles in 2019 and 2021. Ajax is currently top of the Dutch league by four points over rival PSV Eindhoven.
The Dutchman also led Ajax to the Champions League semifinals in 2019, defeating 13-time winner Real Madrid in the round-of-16 before dramatically losing to Pochettino's Tottenham.
Ten Hag will replace Ralf Rangnick who is in interim charge of United until the end of this season.
Rangnick was appointed after former manager Ole Gunnar Solskjær was sacked by the Premier League club in November.
United sit seventh in the Premier League on 51 points with seven games left to play this season and look unlikely to qualify for next season's Champions League.