I am in the U.S. on an H-1B visa, working at a nonprofit research institute as a research scientist. The nonprofit is affiliated with a university. I have been offered a similar job by a U.S. private company. Can I transfer my H-1B visa to work there?
There are two important things you need to consider. The first is the H-1B cap. If your only H-1B employer has been this nonprofit, you have not been counted against the annual H-1B cap, or limited annual allotment. Universities and affiliated nonprofits are not subject to the annual quota for H-1B visas (65,000 base amount; 20,000 additional for U.S. graduate degree holders).
This means that when the private company files an H-1B petition for you, you will need one of the 85,000 visas — and they tend to run out fast. If the quota already has been exhausted for the current federal fiscal year (October 1 to September 30 of the following year), you would not be able to change employers until October 1. Therefore, you'll want to check the H-1B quota usage and evaluate the timing for when the employer files the petition and when you can start the new job.
The second thing to consider is that there really is no such thing as an "H-1B transfer." Each new employer must file a petition to sponsor you, and a prior H-1B approval does not guarantee that the next employer's H-1B petition will be approved.
Your situation also is a great example of how the word "transfer" is not the best use of the word, because it makes no reference to the H-1B cap. Sometimes the word "H-1B transfer" is used to refer to someone who has H-1B status to work at one company and is "transferring" to work at another company. If the H-1B worker already has been counted against the annual quota, then his or her allocation in the prior year's quota is "transferred." So it just means he or she is not subject to annual quota. In your case, you're going from a cap-exempt to a cap-subject employer, so not even an H-1B visa allocation/prior year quota is being transferred.
For more information, see Nolo's articles on "Life in the U.S. on an H-1B Visa" or consult an experienced immigration attorney.