Court documents state the claim against the BBC and BBC Studios has been "discontinued".
The Christmas specials will also air, with the BBC acknowledging it had not been a "straightforward decision".
The former MasterChef presenter is taking legal action against the corporation after he was sacked.
The former MasterChef presenter is seeking up to £10,000 in damages from the BBC, court documents show.
The TV presenter was sacked in July after a report upheld more than 40 allegations against him.
Tim Davie tells MPs he is "not letting anything lie" when it comes to rooting out abuses of power.
Gregg Wallace and John Torode were sacked after an inquiry into conduct on the BBC show.
The Saturday Kitchen host replaces Wallace, who was sacked after a report upheld complaints over his behaviour.
The show's production company says another chef decided not to be included "given recent events".
