The manuscript was penned on both sides of an A5 card for a charity auction nine years ago and sold for £25,000.
The untitled 800-word work was taken from Howard Road, Kings Heath.
Reportedly set three years before Harry Potter's birth, it features the schoolboy wizard's father James Potter, and godfather Sirius Black, when they were teenagers.
They are confronted by two "muggle" policemen after a high-speed motorbike chase but manage to escape on broomsticks.
The manuscript and jewellery were taken during the break-in between 13 and 24 April, West Midlands Police said.
The force is issuing the plea for information almost one month on, in the hope "the appeal goes far and wide among Harry Potter fans throughout the world".