Flanagan has been in charge since 2011 and led the club into the National League North in 2015.
The Nash say the decision was made due to recent poor results.
Curzon are 14th in the table but lost 2-1 to Basford United in the FA Cup following league defeat to Boston United.
“The club regret to announce that due to recent results a decision has been taken to replace John Flanagan as the Curzon Ashton manager,” the club confirmed.
“Over the past 7 years John has done a tremendous job, gaining two promotions resulting in Curzon playing at the highest level in its history.
“We would like to thank John for his hard work over the years and wish him well for the future.”
Curzon are without a game until December 8 when they travel to FC United of Manchester.