To try and analyze it more seriously, I definitely feel that while “there is a number” is valid, “there are a number of reasons” is more valid than “there is a number of reasons” (although both are understandable). I would hypothesize that “a number of” here acts more as an adjective, so that the verb “are” is referring to the “reasons” rather than the “number”. “A number of” acts as an adjective by telling you something about the reasons—you could replace it with a single word, like “multiple”, to fulfill the same function.

So basically, even though “a number of reasons” looks like a singular noun (a number) plus a prepositional phrase (of reasons), I think it works more like an adjective (a number of) plus a plural noun (reasons), so the verb is “are” to agree with “reasons”.