The way baseball free agency works is you have an arbitration period that you have to play out before you can be on the open market. You need to get 6 years of MLB service time before another team can bid for you, but until then your team has exclusive rights to negotiate with you.
If I'm not mistaken, the only way to speed this up is if you get traded, but I could be wrong. If you already know all of that I apologize.
My original team did attempt to sign me, but they low balled me. I received my offer from the Padres about a month into the free agency period. If I had to guess, I'd say around 8-12 teams sent an offer.