You may remain in the United States while a change of status is pending assuming the request was filed prior to the expiration of your current status. However, you cannot begin to work until the change of status request has been approved. There is also a risk of accruing unlawful presence of the change of status request is denied. You should consider speaking to the employer's attorney to make sure it is even feasible to get the H-1B petition filed within the ten days you have remaining in your authorized stay. You can read more about H-1B at http://myattorneyusa.com/work-visas.
Answered on May 20th, 2016 at 3:25 PM