The former Leeds United man joined Bournemouth last week.
The 19-year-old midfielder joined Premier League side Bournemouth last week in a deal which could rise to as much as £10 million, according to The Guardian.
It has been suggested within some sections of the fanabse that the England Under-19 international - who joined Leeds at the age of seven - did not want to leave.
But Monk, who replaced manager Steve Evans earlier this summer, was unequivocal in his claims that the young midfielder did want to leave the Elland Road club.
"Myself and the club wanted Lewis to stay," he told LUTV. "I know the club made huge efforts to keep him here in terms of contract.
"Myself and the club did everything we could to keep him here, but when a player really wants to go, and if it meets the requirements of the club, then it happens. It's unfortunate.
"We'd have loved Lewis to stay here. He would've been an important part of the team, but we move on."
Cook's exit will disappoint the vast majority of Leeds fans regardless of whether he made the decision or not, but his departure perhaps underlines the importance of keeping left-back Charlie Taylor.
The 22-year-old was arguably Leeds' other flagship figure after winning the club's Player of the Year award last season, but losing him would deal United's promotion hopes a huge blow for the campaign ahead.