The corporation thanked the board of Channel 4 for releasing Thompson early. In a statement, it said: "The BBC has agreed to reimburse Channel 4's one-off search fee incurred in its recruitment of a new chief executive."
The BBC said that it would pay a five-figure sum, and that Channel 4 would reciprocate in similar circumstances in the future.
Channel 4 was surprised by Thompson's appointment to the director-general role last month, after its chief executive had publicly vowed that he would turn down any offer from the BBC at the broadcaster's annual general meeting.
Channel 4 chairman Luke Johnson headed negotiations over Thompson's exit from the network after two years as chief executive. He was reported to have been extremely irritated by Thompson's decision to leave and return to the BBC, where he had enjoyed a 23-year career.
The BBC had wanted Thompson on board as soon as possible so that he can play a part in the production of the BBC's Charter Review papers, which go to the government at the end of June.
His other major task upon joining will be to implement the new BBC editorial standards following a review instigated by new chairman Michael Grade. The review was called in the wake of the Hutton Report, which saw Greg Dyke and Gavyn Davies stand down as director-general and chairman respectively.
Sky Networks managing director Dawn Airey is the bookies' favourite to take over from Thompson as chief executive of Channel 4.
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