A statement on the band's website said Salisbury "will complete the October UK tour at least", adding that Brookes would undergo an operation this week.
The group, whose hits include "One To Another," are due to start their European tour in Hatfield on 5 October.
Brookes's illness was diagnosed last week after he collapsed and briefly stopped breathing during a concert in Philadelphia.
The rest of the band's US tour was canceled as Brookes was flown to the UK for treatment.
A spokesperson for The Charlatans said: "The band and Jon would like to thank all well wishers for their kind messages of support."
- Source: BBC