I am in need of a Necchi 538/539 needle plate as shown in picture.

It sounds like there may be an issue with the needle clamp on your Necchi NC204D sewing machine. The fact that the needle falls out when you try to tighten it suggests that the clamp may not be holding the needle securely.
Here are a few steps you can try to fix the problem:

1. Check the needle clamp: Make sure that the needle clamp is tightened properly and that the screw is secure. Also, make sure that the needle is inserted all the way into the clamp.
2. Check the needle: Make sure that you are using the correct type and size of needle for your machine. The needles that came with the machine should be the correct size, but it's good to double-check.
3. Check the needle plate: Make sure that the needle plate is clean and free of any debris or burrs that could prevent the needle from seating properly.
4. Check the needle alignment: Make sure that the needle is aligned properly and that the flat side of the needle is facing the back of the machine.
5. If the problem persists, it's best to contact Necchi customer service or a professional sewing machine technician for further assistance.

It's important to note that working with a sewing machine can be dangerous, if you're not comfortable working with it, it's best to get it serviced by a professional.

Perhaps one of these vendors can help (you'll probably need to email them):

http://www.sewingpartsonline.com/necchi-sewing-machine-parts-model-list.aspx

Necchi Home Sewing Machines www drdanessmh com

Necchi needle plate

You might also look on eBay:

necchi 549 needle plate

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You can use either 16 or 16J. If your threads are loopy on the bottom side of fabric, the problem is either your needle thread tension is too loose or you have a snag in your needle plate where you have broken a needle. Check needle plate and clear snag if any. Then tighten up your needle thread tension until both needle and bobbin threads are in middle of fabric.

There are a number of things that could cause this.
Needle installed with flat in the wrong direction.
Excessive thread tension,top or bottom.
Lack of lubrication. If the machine can do anything more than a straight stitch the mechanism can gum up and not move the needle correctly.
Here is a link to threading directions.
Necchi 3537 Sewing Machine Threading Diagram

Most sewing machines can use a double needle, the one with the single mounting shank and it turns into two needles.
MAKE sure you have Zig Zag needle plate installed to accomodate the double needle.
Turn hand wheel slowly to test then use thread and fabric and turn flywheel by hand to test sew.

Jeff: First it is never wise to put your email address in a public place like this. Second... your pictures didn't come up.

I suspect you have a shim... but not being able to see the pictures. It's hard to advise you.

try sewingpartsonline.com

if the sewing machine is not sewing & staying in one spot the that means the feed dogs are not working properly.
I also have a necchi sewing machine, check the needle plate area that's where the feed dogs are, if the feed dogs are up where you can see them & even feel them with you fingers when you rub your fingers over the needle plate area, if you can't see them good or feel them, then they are not up,,my instruction book shows that you have to remove the extension table, their should be a small door in front of the needle plate area to open that's where the feed dogs are if they are not in the up position, where you can sew, if they are in the down position move the lever to the opposite direction & see if that help the machine to sew, if it make makes no difference which position there in , then it's possible something is wrong with the feed dogs & the machine needs to be serviced

Could be threaded wrong - check manual for proper threading
Needle could be bent - check and replace if needed
Needle plate could be damaged - see if any burrs are on it if so take to service dealer and have removed
Bobbin could be wound wrong - check manual
Bobbin could be bent - if so replace.

The most common reason is the needle not inserted correctly (backwards), other causes can be burrs on the needle plate opening or burrs on the parts in the bobbin area.