Channel 4 signed a three-year deal with Love Productions, which makes the hit baking programme, in September last year.
A BBC spokesperson wished the programme "well for the future".
Paul Hollywood is the only celebrity to be following the show to its new home.
An insider told BBC News: "We don't want to get in the way of them reinventing the programme.
"In this instance, we believe it would be undignified to have one public service broadcaster in a potential dispute with another and the associated costs for each party would ultimately come out of programming budgets.
"We don't believe in driving up costs in the public sector."
Hollywood's fellow judge Mary Berry and hosts Mel Giedroyc and Sue Perkins left the show after the BBC lost the broadcast rights.
Speaking on the red carpet ahead of the National TV Awards on Wednesday night in London, Mary Berry said: "I'm delighted to stay with the BBC and I've got all sorts of exciting projects I'm looking forward to doing."
The TV judge added she would have a new series out in February which would be on "every day".
Channel 4 confirmed the next series of the Great British Bake Off would be broadcast this year.
A BBC spokesperson said: "The BBC is proud of the part it has played in growing and nurturing the programme - doing that is at the core of what the BBC does. We have many exciting projects for the future.  Watch this space."
The show, which attracted millions of viewers, enjoyed seven series on the BBC, beginning on BBC Two in 2010 before moving to BBC One in 2014.
The Great British Bake Off will return to television screens this year, after the BBC waived a legal clause preventing Channel 4 from showing it until 2018.