Only the court with jurisdiction of the child (the last place the child lived for six months or if not six months was born). So only Illinois has jurisdiction and can order paternity.
This is not a comprehensive answer and it is impossible to provide a meaningful response without a consultation.
Illinois as that is where the child lives, unless the mother agrees to one, and then how do you confirm that she used her and the baby's saliva?
My name is Stephen R. Cohen and have practiced since 1974. I practice in Los Angeles and Orange County, CA. These answers do not create an attorney client relationship. My answers may offend I believe in telling the truth, I use common sense as well as the law. Other state's laws may differ.. There are a lot of really good attorneys on this site, I will do limited appearances which are preparation of court documents it is , less expensive. However generally I believe an attorney is better than none, but many will offer a free consultation and a face to face meeting generally will be better, I like my clients to write a short one page history of the fact and questions they have prior to meeting with them, so nothing is forgotten.
Unless the child was conceived in CA, IL has jurisdiction
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