The statement, external said that when the band formed in the early 2000s, it had been faced with a “far more serious medical obstacle that should have ended our dream there and then”.
It said they “point blank refused to accept that we couldn’t do what we’d set out to do”.
They added: “With a wee bit of time we’ll now do what we did then… whatever it takes to get back onstage as quickly as possible, playing music for anyone who loves our thing as much as we do.
“We were blown away by the reaction to this tour and can only promise that we are as disappointed as you at having to wait a little longer to celebrate Costello Music’s 20th birthday.
“We’ll do everything possible to announce the new dates ASAP.”
The Fratellis are set to play at Glasgow’s TRNSMT festival in June.
Costello Music was released in 2006 and peaked at number two on the UK albums chart.
The band toured the record internationally and won the 2007 Brit Award for Best British Breakthrough Act.