Do I need to go to my home country to obtain an invitation letter?

I am an American citizen currently visiting family in Singapore. I have been offered a teaching job in Beijing, and have signed the offer letter, and thus the visa process has begun. However, my company is saying that I need to go to my home country (USA) in order to receive the invitation letter required for a work visa because the Chinese, in the company's words, "Will not issue an invitation letter unless you are a resident of the country you are currently in." Is this true? If it is, what other options do I have? I do not have the resources to go back to America at this. IS there a way around this?