I would offer my educated guess on this. No, not monkeys. Monkeys are not as genetically close to us than other primates. However, I understand the meaning of your question. It would be better if you understand that it the the great apes (For example, Chimpanzees Bonobos, and Gorrillas) that are the closest genetically to humans. Now, if you had asked this question about the great apes, it would have been a much better example (monkeys are not apes). Now, we know that there are distinct species that are nevertheless close genetically and can produce offspring. A male donkey and a female horse produces a mule. A lion and tiger can produce offspring: the ligon. So far, though, it seems that the offspring of distinct species cannot themselves produce offspring. So, can a human and a great ape produce offspring? I am not sure, but I would guess that it is possible. I skimmed the net and found some claims related to this possibility. I have heard of situations though that a monkey impregnated a woman.