I have a trademark registered in Lebanon and I am looking to export to the US. The ® circle (r) is used on my products however we have not registered the TM in the US. Would we face any liability in such a situation since we will be using the circle (r) sign in the US but the trademark is not registered?
Assuming that you are actively shipping to the US and the products are moving through your distribution channels, then you may wish to take steps to prevent misrepresenting or deceiving the consumer. By the way, you can probably just print a small label to cover up your ® and replace it with an ™. That may be adequate. Of course, if this is your situation, you may likely wish to apply for a trademark in the US.
On the other hand, lets say you are selling primarily on eBay from Lebanon. As a result, some customers from the US are ordering your product and you are shipping it to them. Further, let's assume that you do not otherwise have any business operations or affiliates in the US. And, let's assume you are not registered to do business in the US. Then, your potential liability may be quiet low, because, you may be outside the jurisdiction of the US.
Of course, it is always good to consult with a lawyer about your specific situation.
Good luck with your exporting your products.