H1 B Transfer process/suggestion

In Dec 2014 I had changed my job with a well know staffing (company A) firm for a contract to hire job with company B. In 25th Feb 2015 I received a RFE for company A for USCIS. On 02nd March, 2015 I got an offer with company B to get hired full time. I have told company B to get hired full time and they will start their process in first week of May.

Now, I have a prospect offer from company C as the company is excellent to work with respect to Compensation + Benefits + Work Culture. I wish to switch to company C, so my question is can I ask the company C to start my visa process in a middle of RFE which has been received for company A.

Can anybody answer to the question ASAP.

Mostly it should be fine. But you need to check about the RFE reason. If its about you(about Employee), similar RFE may get trigger during any subsequent H1 petitions.