A1 Bro,
I think Even if the the Company wants to hire you, They cannot file for your H1B if you remain in F1(Student and not have the work permit, CPT/OPT) for 2016. You have to prove it to USCIS that you are currently working with the company via CPT/OPT with all the proofs(Paystubs, work details, responsibility etc.) and not the other way. Had that been the case, nobody would go to CPT college seeking for CPT. USCIS have list of process that everyone need to follow to legally work in US. Since you have completed your masters in US and leveraged your OPT I am guessing you might easily find a CPT school and get a CPT and process for next year H1B. Bottom line you have to have CPT/OPT to file for H1B. There is no alternative. The Company CANNOT hire you if you do not have the work permit, that is when the back ground check comes in effect. The company has to verify that you are legally eligible to work in US in other words you have work permit.
I think this is the process Unless Silentblogger thinks otherwise.