Monk, dismissed by Premier League Swansea in December after almost two years in charge, has signed a one-year rolling contract at Elland Road.
But he gave short shrift to the inevitable questions at his first news conference on Thursday about his likely "shelf life" under Cellino, who is on his seventh manager since acquiring Leeds in 2014.
"It isn't my focus. My focus is on the football," he said.
Asked whether the Leeds owner had set him a minimum goal, he replied: "It isn't about that. No-one will ever expect more than me from myself."
But he said his own target was promotion back to the Premier League, where Leeds last played in 2004.
"That is what we are all here for," he said.
Earlier, Monk told the club's website that Leeds's history and reputation -- they have won the English top-flight title three times and were European Cup finalists in 1975 -- influenced his decision to join a team who finished 13th in the Championship last season.
"The passion of the fans, the history, and the ambition of the owner all tie in with what I wanted to do," he said.
The Leeds job is only the second in management for Monk, who became the Premier League's youngest boss at 37 when he became interim player-manager at Swansea in 2014, saving them from relegation.
Monk was rewarded with a three-year contract and led them to a club-best eighth-place finish in his first full season.
But after signing another three-year deal in 2015, he was dismissed in December with Swansea in 15th place following a poor run of form. The club ended the season in 12th.
Italian chairman Massimo Cellino, who bought Leeds in 2014, had a reputation for sacking managers even before joining the northern English side, getting through 36 in 22 years at Italian club Cagliari.
Welcoming his new manager, he said: "Garry is part of a new generation of coaches in England and has a lot of potential to develop, which is something we were targeting for the new season."
(Reporting by Neville Dalton)
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