The UKHSA's Health Protection Team advised that pupils or staff who are vulnerable or immunosuppressed should seek health advice, and state that they "may have been exposed to influenza or another respiratory infection".
The letter said: "As we are currently unable to open, we have implemented a plan for the effective continuation of learning.
"Work will be set in accordance with the usual timetable, where staff are well enough to do so.
"A range of learning activities will be set... any additional teacher support in lessons will take place via Teams message or school email."
The school said students were expected to engage in remote learning on both days if they are well enough to do so.
Staff would be carrying out "safe and well checks" as appropriate, it added.
Congleton High School said it made the decision to close on Wednesday and Thursday due to a number of cases of flu-like illness in the school this week and following discussions with the UKHSA's Health Protection Team and Infection Control Team.