H1B Petition & Visa expired without entering USA

My I-797 petition and stamped H1B visa both have expired now. But I couldn’t enter United States even a single time before it expired. Can I approach another employer to transfer the same petition and assist me with re-stamping of visa without going through the fresh lottery process?

Yes, once you are selected in the cap-subject aka lottery, you are treated cap-exempt for the 6 year time you spend in the US.
In your case a new H1B employer can file a new cap-exempt ( no need for lottery) and once approved, you can go for stamping.

Thanks a lot Kalpesh for the confirmation. Actually, my current employer is not filling the petition stating that it’s not possible as confirmed by their immigration attorney.

Hence, it would be really kind of you and should be of great help if you can share any reference to the official rule applicable to my case which I can point and ask them to check.

Did you share the prior approved copy of I-797 and the visa stamp with your attorney? Are they aware of your past H1B approval?
Your employer should be able to file I-129 based on the prior approval of H1B under cap.