The phrase "to face the axe" means to be in danger of losing your job or facing a negative consequence, such as being fired or replaced. Full definition
They were reportedly facing the axe from the forthcoming season on the X Factor. (longroom.com)
Sales are equally important, and at least one such vehicle has failed to make an impact and is now facing the axe. (topspeed.com)
With the non-judicial foreclosure moratorium passed in 2011, many Hawaii homeowners were given another chance to work out a loan modification with their lender or more time to complete a short sale instead of face the axe of a foreclosure. (bigrockinvestments.com)