No, the GM is ALWAYS the one to broker (and pay) Compels.
Players may trigger a compel by invoking-for-effect, paying a FP (or tag) to the GM, but after that, it is between the GM and the Compelled character's player.
Exactly. While this may seem like semantics, it's an important point because it means the players can never cut the GM out of the compel interactions. A GM always has the right to veto a compel suggested by any player, and the player who triggered the compel has no further say in the negotiation of the compel. Which if your players have a surplus of FP, I would highly suggest being very strict on the compels until they start sing the fate points.
It s a blind suggestion, but to the OP I would say you way want to curb how many fate points you're handing out. FP surplus means either you create a baloon effect in your game, or the players steamroll everything.
My personal GM method is to generally keep the players below their FP refresh level by the end of the session. If they want to save up FP they have to really hang on to them.