The Danish coach took over at the Liberty Stadium in June 2012, replacing Brendan Rodgers, and guided the side to a ninth-place finish and lifted the Capital One Cup - the first major silverware in the Welsh club's history.
However, the Swans have struggled this season, only winning one of their last 10 Premier League fixtures, and have slipped down the table, now just two points above the drop zone.
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“It is a decision we have taken reluctantly,’’ Jenkins told the club's official website. “But it’s a decision made in the best interests of Swansea City Football Club and our supporters.
“It is the first time in nearly 10 years that the club has parted with a manager in this way, but we had to remove the constant uncertainty surrounding the club and Michael’s long-term future with us.
“I had a meeting with Michael today in a final attempt to support him and establish a way to improve the work of the backroom team to secure the results we need over the final 14 Premier League games.
“However, after thinking long and hard about the best way forward, I felt it was unlikely we would achieve a stable environment at the club to allow us to get back to basics and produce the performance levels that have served Swansea City so well over the last few years.
“Now we need to put that uncertainty behind us and move forward as a united football club on all fronts, while placing on record our gratitude to Michael for the work he has done over the last 18 months and wish him well for the future.
“I hope all our supporters can fully understand how difficult this period has been for us and I would urge everyone connected to the football club to get behind Garry Monk, the staff and players."
Monk's first match in charge will be the south Wales derby against rivals and fellow relegation contenders Cardiff City, who won the reverse fixture 1-0, while the club face a Europa League last-32 tie against Napoli later this month.