The BBC said it would be "undignified" to keep one of the most popular shows on UK TV away from viewers, the Guardian reported.
An insider at the BBC told the paper: "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."
The news will be highly welcome to Channel 4 after it was seen to have taken a significant risk when it bid £25m a year to acquire the show, which has been on BBC One since 2010.
The move was questioned by some commentators after three of the four stars of the show announced they would not be moving to Channel 4, raising questions about its ability to retain the huge viewer numbers it has commanded in the last few seasons.
A BBC spokesman said: ""We wish the programme well for the future and look forward to seeing who is cast on the show.
"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."