It seems certain that the number of monosyllabic English words is somewhere between 3000 and 40,000. If you exclude inflected forms (see below), I'm fairly certain that the number would be below 30,000.
Past at certain point, you have to decide on a clear definition of "word" in order to get an accurate answer:
probably the most important thing is whether you count inflected forms like walks and walked as separate words from the word walk; because so many English words can inflect (nouns to make plural forms, verbs to make the past-tense form or the third-person present tense form) counting these as three words rather than one will probably substantially increase the number of monosyllabic words.
Many infrequent dictionary entries are proper nouns; you need to decide which, if any, of these are included in your definition of "English words".
Presumably, unrelated homophones (like steel and steal) will be counted separately. I would also count unrelated homonyms (like bat "flying mammal" and bat "club") separately. It may be a bit less clear whether related words that are homophones and homographs, like walk (n.) and walk (v.), or moved (v.) and moved (adj.), should all be counted separately.
There are probably fewer than 15,831, since Chris Barker gives that as an estimate for the total number of distinct syllables found in any position in real English words ("How many syllables does English have?", link to archived version found in Andrew J. Lintz's answer to the question How many syllables are in the English language?).
The abstract of "Sensory experience ratings for over 5,000 mono- and disyllabic words", by Barbara J. Juhasz and Melvin J. Yap, refers to "2,857 monosyllabic words used in the Juhasz et al. study" on "sensory experience ratings". I would infer that for a word to be useful in this kind of study, it would have to be reasonable to expect an English speaker to be familiar with it, so I think that we can say that there are definitely at least 2,857 monosyllabic words that are fairly well established in English vocabulary. The list of words is available to download in the "Supplementary material" section; I glanced at it, and it does not seem to contain inflected forms ending in -s or -ed.
The Wiktionary category "English 1-syllable words" has 8,671 pages, but there are certainly erroneous inclusions; e.g. 50 or so suffixes (like -ous) and some polysyllabic terms where the pronunciation has only been entered for only part (e.g. slough of despond). It also includes a number of proper nouns and obscure acronyms like BRUK, which you might not want to include in a list of "words". So it's not a hard lower limit, but it might give you an approximate idea of how many single syllables might be considered by some definitions to be English words.
Patrick Corliss wrote an answer to the earlier question "Is there a list of syllables contained in US English?" saying that he had found over 10,000 single syllable words (including names, adopted foreign words, and inflected forms of words).