H1B Transfer Please Help!

My first three years' H1B expired on July 3. My current employer filed the renewal quite a while ago but still haven't gotten the approval notice yet. Got a job offer from another company. Will I have trouble getting the H1B transfer done given the fact that my current H1B approval has expired and the extension hasn't been approved?

Contacted HR in the new company but haven't gotten a response. Please let me know if you know any situation like this. Many thanks!!