The Old Trafford giants confirmed that the Ajax manager would take the reins in the summer on a three-year deal, with the option of a further season last month.
Ten Hag is the permanent successor to Ole Gunnar Solskjaer and planning is under way ahead of his arrival, with Ralf Rangnick's underwhelming interim stint coming to a close in the May 22 season finale at Crystal Palace.
Preparations intensified the day after Ten Hag oversaw Ajax's Eredivisie title triumph with a 5-0 win against Heerenveen as Murtough watched United beat Nottingham Forest in the FA Youth Cup at Old Trafford.
United's football director Murtough is understood to have flown out to Amsterdam to see the Dutchman on Thursday morning.
While the pair have been in dialogue since the appointment, intensive discussions over transfers and strategy were parked until Ajax had won the title.
It has been reported that ex-England boss and former United coach Steve McClaren, who worked with Ten Hag at Twente, and assistant Mitchell van der Gaag were also in attendance.
There is plenty of work required to turn things around at Old Trafford, with midfield among the key areas requiring improvement this summer.
Barcelona's Frenkie de Jong, who rose to stardom under Ten Hag, is among those admired by United but a deal currently looks a way off.
Meanwhile, Ajax confirmed on Thursday afternoon that Club Brugge's Alfred Schreuder will succeed Ten Hag as head coach in the summer.