I would wash the fabric with Retayne. Some quilt shops carry it; otherwise it is easy to order online. Retayne is designed specifically to set unset dyes.
Bleeding can be due to dye not being set. However, it can also be due to excess dye in the fabric. Since you've soaked it twice and it is still bleeding, I would tend to think the problem is unset dye.
Not all fabrics will absorb unset dyes equally. The white you are using may be resistant to dye bleeds. However, other fabrics you use in a quilt may not be resistant to this bleed.
The safest approach is to wash and dry with Retayne. Some fabrics need two Retayne treatments. If the fabric still bleeds after that, it needs to be tossed or used for something that won't be washed or doesn't involve other fabrics. The only alternative to Retayne is to test the fabric against *every* other fabric used in the quilt (not something I would want to do).