Travel to USA without transferring petition from current employer??

[SAN1988]

Hi,
I have valid stamped H1-B visa from my current employer A who is not sending me to USA.
My new employer basically a consultancy firm(B), saying me that they will initiate the invitation letter / offer letter from their company for me.
With those documents and approved petition from A, I can travel to USA, and once I landed to USA, they will initiate the process
for SSN and transfer of petition. Is it legal/possible to travel without transferring employer A's petition to new employer??.
Because I will have to resign from employer A in order to travel to USA.
The suspecious thing is, B is saying that at POE, I have to tell them that, I am going from my current employer i.e. A to work with my client which
is B.

Can you please help me with this. It's very confusing and seem illegal to me.?

[daggit]

There has been an increasing amount of POE scrutiny of late with certain visa types. Shady consultancies is one aspect, a good number of which are US-based desi-owned. You may wish to sort things out with your employer first.. before traveling to the US and facing additional scrutiny by the CBP.