If I offer a contract, then I see another club has, later, offered a contract, I go back and up my offer by about 20%, so if I'm offering £20k/week, I would up that to £24k/week, same with sign on fee, agent fee, bonuses.
Now, if they already have a contract offer from another club, and I'm offering second, I ask what they want. Generally, I meet that, and add one small bonus. Perhaps I add something like a bonus for international caps.
Most of the time, this works. However I had two players escape me when I first took over as Man City manager. I came from Brighton, so a much smaller club, and I think the players would not sign for me as I was not so well known and they signed, instead, for Man United and Real Madrid which had very established managers, despite my using the tricks, above.