You need to translate the sentence in your head before you decide which answer you're going to pick (if you're not entirely sure).
First read it without the gap filled in and try to guess what it wants to say.
The example sentence here approximately wants to say "It looks as if it's going to rain tomorrow."
The information of "looks as if" is already contained in "雨が降りそう" but as it happens, in your example they want to turn this general (objective) sentence into a subjective sentence. This information is contained in the "~がする" at the end of the sentence. There are many examples of expressions of personal feelings in combination with "~がする", for example:
and many more, in particular ~予感がする.
Now let's see why 予報 doesn't work here. As you pointed out, 予報 means "(weather) forecast". Using this in the gap, the sentence would roughly translate as follows:
"The forecast makes as if it was going to rain tomorrow."
Note that I purposefully translated it as closely to the original as possible with the result of it being ungrammatical. This is because the "~がする" at the end points you to a personal feeling which then used with "weather forecast" ends up not fitting together because "having" a forecast is not a personal feeling.
So to more precisely answer your question:
Is the objective version and translates as "It looks as if it's going to rain tomorrow."
Is the subjective version and translates as "I think it looks as if it's going to rain tomorrow."
~そう can always be used to say "looks as if". For example
Hope this helps.