If a company finds the right person for a job, they are willing to pay more at that point rather than start the search/hire process all over again. Pick a number and say 'if you can get to this number it is a no-brainer for me and I'm in'. More often than not, they will do it. Saves them money and effort in the short term and possibly long term.

Once you sign on the dotted line, you lose a lot of leverage and you only get it back if they want to move you into a specific position and don't want to hire from the outside.

Good luck!