$20M for 5 years with a $10M signing bonus and player option is the max contract. Elite players will go for that. If two or more people have offered max (or equal) contracts, tiebreakers are used to determine who has the top bid. Tiebreakers can be found in FAQ, or a search of the forums. If you have the top bid it will show up as a 1 in a bubble next to the guy's name. If your bubble doesn't have the 1, you don't have the top bid.
There is no telling what a bidding war might produce. I have seen guys go for $12M for 5 years who started out at $5M for 3. There are also some guys that will start at $6M and go for $6M or less if their value drops. It really can vary from league to league, so it's hard to say how your league operates.
One thing to take notice of is whether a player is a Type A free agent or not. If you sign a Type A free agent, the previous franchise will be awarded your first round pick if it is above 16, or your second round pick if you are picking 1-16. A lot of teams don't want to give up picks, so even though a Type A might be a better player than a Type B guy, the Type A guy will get less money, or not signed at all (until draft prospects come out at which point you no longer lose a pick). Draft picks/International signings are key to building your franchise long term, and there is little talent (usually) below the first round.