I have applied already applied for my work permit extension, but in the present situation I need the new work permit immediately to obtain US visa. I will be traveling in July end, and I applied for extension of work permit on May 25. Is there any way I can request a quick processing of my extension application so as to receive the new work permit by June end? Whom can i contact in this case?

There is no mechanism for expedited processing of existing cases for inland processing. You can try reaching the call centre at 1-888-242-2100 however.