Ten Hag, who is presently the manager of Ajax, will take over from Ralf Rangnick, who is currently the temporary manager of Man United, at the end of the season.
Man United's leadership is currently hammering out the finer points of the agreement, with Ten Hag already knowing when he would be appointed.
The Red Devils are resolved to show respect for Ajax, who are still in contention for two trophies, and will announce Ten Hag after the Dutch Cup final to avoid any distractions that could jeopardize their campaign.
The Premier League giants have traditionally had good relations with Ajax, particularly with chief executive Edwin van der Sar, a former Manchester United player who wants to keep such ties going.
With only five games left in the season, Ajax will face PSV Eindhoven in the KNVB Cup on Sunday. They are four points clear at the top of the Eredivisie table.
An announcement is expected before Ajax clinches the title, but not before the Cup final on April 17, with Man United still hoping to clinch the Ten Hag by the end of the month.