Liverpool paid the most agent fees last year.
There's a reason for that. They apparently allow agents to get involved with negotiations for players exiting sponsorship deals and entering new ones when leaving a previous club and joining Liverpool.
eg.
Club have deal with Nivea Men. New signing has personal deal with Gillette courtesy of his agent. Club negotiate the buy out with agent, who buys him out of their contract with Gillette, before signing.
The major issue nowadays is players have multiple personal contracts and the issues with some transfers is getting them out of these contracts beforehand. If they're a globally renowned player, the fee is obviously bigger.
But apparently Liverpool don't deal directly with the player sponsors but the agent does that on their behalf.