H1b application for rejected asylum

At this time, an employer can not file a FORM I-129H if the position is subject to the quota. However, if the job position that you are contemplating is cap exempt, then your prospective employer can file the FORM I-129H. You may be able to adjust status to a cap exempt H-1B if you have maintained yourself in proper F-1 status.

A rejected asylum application should not necessarily prevent other, valid applications from being approved. However, before you give up on asylum, have you already appealed that decision? I suggest you meet with a local immigration attorney to discuss your best options at this point.

You may seek a genuine H1B application initiated by a valid sponsor, provided the job offer is genuine and not intended to bootstrap your desire to stay in the United States under any pretext, given your prior procedural immigration history.

You are also eligible for OPT before you apply for H1B. It may be best to have an attorney assist you.