I've had a few team members that had offers to leave, only one of which I countered...and he didn't ask me to, that was my choice. That was a rare occurrence, an old boss in a new gig called him out of the blue (he wasn't looking), he was a rockstar on my team and was going to get promoted in a couple months anyway.

A few of the others hinted at wanting a counter and I responded by shaking their hand, wishing them well in their new gig and starting the transition. One of them came back the next day and asked to rescind her letter as she decided that the other offer wasn't what she thought it was. I gladly kept her at her current salary and she was the same solid if not spectacular performer ongoing. Never did ask her if she really had an offer.

If you have another offer that you would accept but would stay for more money, can't hurt to let that be known, but I sure as heck wouldn't count on it no matter how important you are...or think you are.