I have been issued notice of intent to deny from asylum office and my I-94 is still valid, has my clock started?

If your admission has not expired yet, a final denial of your asylum application should not trigger removal proceedings against you. But USCIS seldom bothers to not do something only because it makes no sense or is downright wrong. So it is possible that the asylum office will issue a Notice to Appear together with a denial of asylum, commencing removal proceedings; and there is little you can do to prevent that. Overstaying your admission might be of somewhat lesser importance in your case. What is the worst thing that can happen if you overstay? You can be denied admission to the U.S. in the future, or even be barred from entry to the U.S. for 3 or 10 years (if you overstay 180 days or 1 year, respectively). Well, if your asylum application gets denied, your entering the U.S. in the future will be unlikely - regardless of your compliance with the dates of your current admission. First, by entering the U.S. on a non-immigrant visa and then asking for an asylum proves that you lied to the U.S. consul when you applied for a visa; it is called visa fraud, and it makes you unwelcome in the U.S. Second, by filing an asylum application, you stated that you cannot go back to your country out of well-founded fear for your safety, well-being, and even life. If you do go back and later come to the U.S. consul asking for a visa, you have to prove that your ties to your own country are so strong that they, essentially, guarantee that you will return home. Well, to a reasonably-minded consular officer, there is a rather glaring contradiction between your claim of fear that bars your return to your country and your claim of the ties to your country that will compel your return there... The bottom line is that, if your asylum will be denied, you'll have little chance of getting into the U.S. on a visa again, regardless of whether you overstay your admission or not. As to your last question, it is absolutely impossible to answer without thoroughly knowing your case. Sometimes, USCIS Asylum Office can change its initial perception of a case and, after reading a rebuttal, recommends a grant of asylum. After all, the very fact that the officer issued a notice of intent to deny, instead of an outright referral of the case to the Immigration Court, would seem to indicate that the officer has doubts about the merits of your asylum claim and wants to give you an extra opportunity to prove your case. How likely it is to happen, I cannot say without reading the materials submitted to USCIS, the notice of intent to deny, and the rebuttal. If Asylum Office does not approve your asylum application, it cannot deny it - it has to refer it to the Immigration Judge, and you would have another chance to present your case. If the judge denies you asylum, it is going to happen 3-4 years from now. If you appeal the denial to the Board of Immigration Appeals, the denial will not become final for, at least, another 2-3 years. By that time, your I-94 will be long expired. So it might make sense to stop worrying about your I-94 expiration and concentrate on presenting to USCIS the strongest rebuttal you can manage.

If you are denied, your case likely be referred to the Immigration Judge, where you will go through a series of hearings on your asylum application again. If you fail there, you will be deported. In the immigration Court, you will be given additional time and permit to work after the certain time. You will not be in violation of your status, until the decision is made in the Court. Those proceedings are called removal proceedings. If you fail, you will be removed from the Country, unless you decide to voluntarily depart at any time of the removal. You need to review your case with an immigration attorney to see if anything could be done to help you. Apparently, they did not believe you if they intend to deny your application.