"in the coming days" is correct and usable in written English. You can use it when referring to future events that are likely to take place soon. For example, "We will be gathering more information in the coming days to determine our next steps.".
In the coming days, a story emerged.
Announcements will follow in the coming days".
They will report in the coming days.
Further such murmurings are expected in the coming days.
In the coming days, we shall assuredly find out more.
Eskinder plans to appeal his sentence in the coming days.
Both features will be rolled out in the coming days.
Being a terminologist, I care about word choice. Ludwig simply helps me pick the best words for any translation. Five stars!
Maria Pia Montoro
Terminologist and Q/A Analyst @ Translation Centre for the Bodies of the European Union