The 66-year-old Scotsman and former striker for Ipswich Town and the national side has been absent from the talkSPORT breakfast show in recent weeks after he was taken ill on air.
Brazil played for the Blues from 1977 until 1983, scoring 80 goals.
He addressed his radio absence in a post on X on Monday morning.
“Hi Guys, I hope that everyone’s well,” Brazil said.
“Some of you will have noticed that I’ve not been around.
“I recently had a life-saving operation and I’m deeply grateful to the brilliant medical team who have looked after me.
“It will be some time yet before I’m fully back to myself and for now I am focussed on my recovery in private.
“Thank you to my family and those close to me who’ve been supporting me/ putting up with me!
“I’ll see you all soon, AL.”
Brazil has hosted his morning show for more than 25 years.