Here is my Scenario:

I am working with company 'A' now on H1b status and I got a offer to work as an full time employee for 'x' company. But company 'x' asking me to file my h1 transfer and they would help me on all paper works to file my H1b Transfer.

As far as I know, only employer should sponsor and file for H1 Transfer. How can I approach to the employer? and Is it legal to file H1 Transfer by employee(myself)? please advise.