If your visa petition is pending at a U.S. Consulate, you could come to the U.S. on ESTA (or B1/B2 visa), as long as the activities you engage in the U.S. are allowed under the ESTA/B visa rules. However, the CBP officer at the border could question you about the activities as he/she may suspect unauthorized work.
You were originally granted an O-1 visa in March 2018, and your O-1 visa recently expired. (before that, you were in the U.S. on an E-2 visa for about 8 years). You had to leave the U.S. and you are currently in Europe. Your renewal petition was approved by USCIS and you filed a Consular application already. Your visa application is pending at the Consulate and you would like to come to the U.S. on ESTA for couple of weeks. Is this allowed?
You filed your E-2 initial investor visa application in December 2020. The application is still pending at the Consulate and you have not been able to schedule an interview at a U.S. Consulate. You want to come to the U.S. for couple of weeks on ESTA. Is this allowed?
What is the purpose of your travel?
In both scenarios above, the answer to whether or not you can come to the U.S. will depend on what is the purpose of your visit. If you want to come to the U.S. to engage in activities that are allowed on ESTA or B-1or B-2 visa, you could come to the U.S., but the CBP officer at the border could suspect that you want to come to the U.S. and work while you wait for your visa appointment.
In the first example above, the officer could suspect that you want to keep working for you O-1 employer while coming here on ESTA. In the second example, the officer could suspect that you may want to start running the E-2 company before the E2 visa is approved.
As mentioned above, as long as you want to engage in the activities that are allowed under the appropriate visa category, you will not violate the ESTA/B visa status but you should be ready to explain why you are coming to the U.S. and what you plan to do (e.g. activities such as visiting your friends, travelling, moving out from your apartment, etc. would be allowed). You want to make it clear to the CBP officer that you are only coming to engage in the allowed activities and you will not work for your past/future employer.
Additionally, you should bring evidence that supports the activities you will be doing in the U.S. with you when entering the U.S. (e.g. evidence of hotel bookings/flight tickets if you are travelling, evidence that your lease is up if you will be moving out etc.). You should have this handy in case the officer asks to see the supporting evidence for your activities.
To sum up: while there is nothing in the immigration regulations that would prohibit you from entering the U.S. on ESTA/B visa while your visa petition is pending, you may face some extra scrutiny by the CBP officers as the border (especially if you have been living in the U.S. and working here for several years).
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