では in this sentence is a collocation of the ren'youkei of だ (which is で) + は, which expresses that something is the premise of a declaration/judgment [1].
So Xでは here means "according to X" or "based on X", i.e.
4日に出した予想では、今年一番早く咲くのは福岡市と高知市で、今月13日です。
According to the prediction given on the 4th, the earliest they will bloom this year is in the cities of Fukuoka and Kochi, on the 13th of this month.
Another example:
天気予報では明日は晴れるそうです。
According to the weather forecast, it will be sunny tomorrow.
You can also substitute だと for では in either of these sentences; the meaning is the same.
天気予報だと明日は晴れるそうです。
According to the weather forecast, it will be sunny tomorrow.
[1] デジタル大辞泉 (dictionary.goo.ne.jp)
で‐は
[連語]
➊《断定の助動詞「だ」の連用形+係助詞「は」》判断の前提を表す。…であるとすれば。…だと。「雨では中止になる」「彼ではだれも承知しないだろう」