Your question calls for total speculation on behalf of any Avvo attorney. There is no "duty" to disclose a job opening.
The above is general legal and business analysis. It is not "legal advice" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.
A budgeted new position does not mean that the position is currently open. Plus, they may already have someone in mind. There is hardly ever a requirement for a private company to publicize new positions.
With that being said, you may have a leg-up on the competition if they don't already have someone in mind. Play it cool and work hard. Don't question their motives to them. Who knows, maybe you'll move yourself into first position.
The above statements are provided as general information and not intended as legal advice. Each matter has its own set of unique circumstances that cannot be adequately addressed without consultation. You are strongly advised to hire an attorney licensed to practice law in your state to represent you.
I think the only chance you have of asserting a cause of action is if you are in a legally protected category (race, gender, nationality, age, etc.) and can show that they are discriminating against you based upon your protected status.