Ultimately the reason is the remnants of verbal morphology in Old Chinese. (Recently every time I post on japanese.SE it seems to be about Old Chinese...) Modern Chinese is a very low-morphology language, but reconstructed Old Chinese had some derivational morphology which is responsible for noun/verb pairs with different readings on the same character (for instance, 長 has a verbal reading zhang3 and an adjectival reading chang2).
Most reconstructions of Old Chinese posit a morpheme *-s which, among its functions, nominalises a verb. Just as 塞 has the two readings サイ and ソク in Japanese, it has two readings coi3 and sak1 in Cantonese (I use Cantonese because, unlike Mandarin, it preserves the syllable final -k which was still present at the time the reading was borrowed into Japanese).
Baxter's reconstruction of the verb 塞 is *s?ək, and the nominalised form (understood as a built up thing) with the nominalising suffix is *s?ək-s. Middle Chinese sound changes caused the verb to retain its coda, while in the form with the suffix, the coda became a glide. As such, when these were borrowed into Japanese, they already existed as a noun/verb pair differing in the coda.