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If you can remove the linen from the furniture such as cushion cover, put it into a cold water bath and swish gently. Hang to dry. If it can't be removed, try cold water in a spray bottle. Spray the fabric well and blot it up with a clean dry towel. May have to repeat. I also like the product Folex for cleaning upholstery when there is actually dirty spots.
Are you sure the upholstery is actually real linen? Not a common use for the fabric. There are look-alike fabrics that are much sturdier.
Lestoil Soak in lestoil and then wash with regular detergent with some extra lestoil in it.
Have you tried woolite spray
Take a look at these links and see if they are helpful to you! Good luck on your project. Happy New Year to you!
https://www.bing.com/videos/search?q=How+do+I+remove+water+stains+from+linen+upholstery+fabric%3f&qpvt=How+do+I+remove+water+stains+from+linen+upholstery+fabric%3f&FORM=VDRE
https://www.hunker.com/13423871/how-to-remove-water-stains-from-fabric
I have an ottoman in a microfiber material - we actually use it for our feet when sofa surfing. I like to clean it regularly with my little carpet devil spot cleaning machine. The first time I only did the spot I was trying to remove. When it dried it looked horrible and you could see the water stain from the cleaning effort. I thought it was ruined but decided to just water stain the whole thing. I looked great after that. So now I just clean every inch if I am going to clean any. I also found that FAST drying helped minimize any signs of water stain, so I turn on the gas fireplace and put the ottoman right in front to dry as fast as possible.