If someone comes to you and says something like:
I’ve found a new job, it pays me X% more. I’m going to leave unless you can beat it.
Let them go. With a happy heart and a sincere smile.
They’ve come to the end of their time, although they may not have realised it yet and are trying to cling on. It’s the natural end of things and it should be accepted.
If they’re angling for more money they are fighting yesterday’s war. They have probably asked for more money and you’ve been unable to match their demands, either because it’s not viable or they’re not worth it.
Or perhaps you didn’t listen when they told you or perhaps you didn’t fight hard enough for them when it came to it. If, as a manager, you are surprised by this turn of events then you need to learn from this.
Either way, my advice will be: let them go.
Why? Because at this point, however you’ve got here, it’s now about the money. And as most of us in knowledge based work already know, money is a terrible motivator.
If you did engage in financial negotiation it will likely lead to something less than their current job offer, whether real or in their heads. Even if you could resolve the issue, there are always other fundamental reasons why someone is looking to leave. A low salary, perceived or otherwise, is how they measure their unhappiness. But the root cause is elsewhere and it’s probably fundamental and unsolvable, at least not with expending significant management time.
In almost all situations, let the person go onward in their career. Everyone is replaceable, including you! Or at least, they should be.