I have a question regarding the process and concerns of a H1B transfer.

I am in a job on an H1B visa and received an offer from another company. Talking to the HR person and pointing out that I would like to get a counter-offer from my current company, she advised against it saying that "they might fire you" and you would be out of status until your H1B transfer is approved. However, looking at online resources I read that I could start as soon as the new company files the H1B transfer (and receives the receipt). This should not take weeks I assume.

Does anyone have a recommendation regarding the approach I should take? I want to get a counter offer (since I am also considering to stay), but do not want to run into immigration issues.