I always assume embroidery thread will bleed until proven otherwise. Especially red thread. The threads with the least bleeding are DMC and Sajou. Sajou is guaranteed not to bleed. You can order it online. I use DMC even through some colors sometimes bleed. Unlike Sajou, DMC makes perle thread, which is usually used for crochet, and is used for Palestinian embroidery. I like using perle for cross stitch, because it is a single unplyable strand, good for use on adia cloth, and where one wants the cross stitches to look like a thick squarish bump, rather than a distinguishable cross.
DMC red perle sometimes bleeds in the wash. So if it is used on something that is going to be washed, or a really nice piece that you will want to care for properly, the piece will require post stitching, prefinishing treatment.
This is a piece I just finished stitching. It still needs to be finished. Its a Ramallah style veil. It’s huge and took me 4.5 years to stitch. Its mostly red DMC on white. Cleaning this during the finishing process has the potential to be hazardous to the piece. This is how I dealt with it.
Spot clean with dish soap any apparent stains.
Soak in bath of 1 part white vinegar to 1 part water until dye stops coming off the textile.
Wash in dish soap and cold water. This might release more dye.
Rinse in the warmest water you might use to wash the piece, or temperature of the warmest humid weather to which it might be exposed. I use luke warm water. This might release more dye.
If dye continues to be released, repeat 2-4 only changing the vinegar solution if it becomes substantially discolored. If no more dye is released, soak in the vinegar solution for a couple more hours, rinse out thoroughly, and dry.
Here, I was fortunate. I had next to no bleeding from this piece. However, I was careful and went through all the loose color removal steps because I have had incidents with red DMC in the past, and because I do not want to mess up this piece.
WARNING: Do not hang up a piece to dry until all the loose dye is out. If you do, the dye will migrate down the piece leaving permanent streaks. The red ‘removable’ iron-on pencil will sometimes run as well. If you have streaks on a dried piece, as a last resort, you can spray on Windex, wait 5 minuets, then do the vinegar wash thing again, in an attempt to remove the streaks. Windex is a miracle worker.
Also, what ever you do, do not put the piece in the dryer if you have a bleeding problem, or if there are still stains on it. The dryer will bake in the color and you will have ruined your piece. You should not be putting anything embroidered except dishtowels in the dryer anyway.
Alternatively, you can wash the thread before you stitch, but that is a serious pain.