there is perhaps some historical connection between the く sound and い sound, either phonologically or semantically.
I think the answer from blutorange addresses this.
Maybe these two classes of words [〜い adjectives and 〜く verbs] diverged from the same class of words somehow?
I'll disagree with blutorange about this part, as his answer is (I believe) ...
According to Morohashi (大漢和辞典), the original Literary Sinitic meaning of the compound word, 勉強 [m-jenX g-jangX] are indeed different: "1. Strive with all effort. 2. Put effort in study. 3. Do business with small profit margin. 4. Section from the book 顔氏家訓. 5. (modern Mandarin) force; compel."
In fact, definitions like this are closer to the modern Japanese ...
Yes, for example consider the beginning of 枕草子【まくらのそうし】:
If you rewrite the adjective and verb forms to their modern form:
Observe that an adjective such as 美しい was originally 美しき (before a noun, 美し at the end of a sentence), with a shift of き→い, see イ音便. 咲いて was originally 咲きて, where you can ...