What I would try, first - Lay the block on a white washcloth or towel, then just blot with a wet/baby wipe (use the plain ones, NOT the ones with lotion). Don't rub; just keep blotting. Change the cloth/towel, when it gets too wet. Try not to wet the embroidery thread, if possible.
Believe it or not, wet/baby wipes will safely remove many stains!