Though I dont know the correct answer, I would imagine there would have to be some sort of pre-FA interest a player has for a team that makes them sign to quickly.

What im thinking as far as the Lions go, is that FA's will probably have more interest of coming to Detroit now that they are a legitimate contender.

I think what is bolded is how it goes down most of the time. The coach or whomever is making the offer has a deal with some rough numbers (We'll give you a five year deal worth $40M) and then the player can say 'sure, that sounds like what I'm looking for'. At that point they've come to an agreement, and thereafter it's up to the front office and the players agent to finalize the deal and get the pen on paper.

Remember, when free agency starts there's no action going on at the practice facilities, so they don't have to be under contract to feel safe to practice or anything like that. They just come to an agreement, then once the deal is finalized it has to be submitted to the league office for approval. That's the final rubber stamp on making it valid.

I think it is more likely that the team and the players agent talk well before FA starts. Teams can't talk to the Players. But, I don't think anything says they can't talk to the agents. Especially, if they have a payer on their team with the same agent as a FA they are interested in.