As a visiting scholar, my J1 visa will expire at the end of this month. A company wants to hire me as a research engineer, and has applied the H1B visa for me to change the status. Due to the H1B visa yearly gap, the H1B application is pending, and I can work for the company only after October 1, 2013. Do you think I may have the status problem after the J1 visa expiration and before the H1B approval? 

To qualify for the H-1B visa approval to change your status from J1 to H-1B in the United States, you need to keep a legal nonimmigrant status on October 1, 2013. Since your J1 visa will expire at the end of this month, USCIS may approve the company's H-1B petition for an alien's worker, but you may not be able to change your status from J1 to H-1B in the United States.

If that situation happens, after the H-1B petition is approved for the employer, you need to travel to your home country to get the H-1B visa on your passport at an U.S. Embassy, and then return to U.S. to work for this company.