The 38-year-old was expected to extend his stay with the Yorkshire club, but talks with owner Andrea Radrizzani broke down after the Italian completed his full takeover.
The former Dutch worldwide defender joined the Championship outfit last summer signing a two-year deal with the job being his first senior managerial role, with his previous jobs - firstly at PEC Zwolle and then in Amsterdam with Ajax - having seen him take more of a backroom role.
"Speaking to the chairman Steve (Gibson) especially and all the guys at the club, it was that ambition and challenge that ticked all the boxes that I was looking for".
"It's important to make clear in terms that there are two separate situations", Monk said.
He left the club following a takeover by Italian Andrea Radrizzani, and claims the new ownership model and structure of the club is the reason why he made a decision to depart. There's no getting away from that.
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Middlesbrough were relegated from the Premier League last season following promotion from the Championship the season before. I'm ambitious, I want to be working and I want to find the right challenge. "It's a competitive and hard league but this club is equipped and we're all determined to do it".
Stam's agent Henk Timmer has essentially ruled out any possibility of a move to Leeds, insisting it would take Premier League interest to consider a departure.
He also accused Monk of using negative tactics, and it seems Leeds fans do not want Stam due to his own defensive philosophy.
"That's how I work best".
Monk will get on with appointing his backroom staff, with James Beattie likely to join from Leeds.