1. Set a “bribe value” for the NPC.
You could put together a set of charts based on income, wealth, sense of duty, and/or potential consequences. But mostly you can just make a judgment call on “that sounds like about the level where this gal might be influenced by the offer of money”.
2. If a bribe equal to the NPC’s bribe value is given, grant a +1 circumstance bonus on associated checks.
Keep this in mind when setting the “bribe value”. It’s not the value at which the NPC immediately buckles like a well-worn belt. It’s the value at which they think, “OK. That’s slightly tempting.”
3. For each doubling of the bribe value, grant an additional +1 circumstance bonus.
So if the bribe value of the NPC is 1 gp, then a bribe of 2 gp gives you a +2 bonus; 4 gp gives you +3; 8 gp gives you +4; and so forth.
There’s no functional limit to the size of the bribe the PCs can offer, but obviously this exponential system does provide a practical limit. Start with a bribe value of 10 gp and you’d need a bribe of 10 billion gold pieces to get a +10 bonus. (Again: This is another reason to keep the initial bribe value at the “slightly tempting” level and not the “likely to accept this bribe” level.)
Note: The system assumes that you don’t tell the players what the bribe value of the NPC is. (Although maybe they can find that out through Gather Information or Sense Motive checks.) With that in mind, use some common sense and rounding while adjudicating the system. For example, pure math might say that you need to bribe 256 gp to get a +9 bonus and 512 gp to get a +10 bonus. If a PC bribes 250 gp, though, they should get the +9 bonus and if they bribe 500 gp they should get the +10 bonus. This is all meant to be an easy-to-use mental guideline for your benefit, not a binding contract written in blood.
Keep in mind that offering a bribe is risky: If the resulting Diplomacy check is blown even with the bribe in play, the target may become insulted or angry in response.