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The vacancy arose when Adrian Chiles decided to leave the BBC to become the main anchor on GMTV.
Previous reports have suggested that Castle's future on the morning show is in doubt and the Daily Mail claims that he is now planning a move to the BBC, where he already hosts the coverage of Wimbledon.
"Andrew's always enjoyed working for the BBC during Wimbledon and they know him very well from it," a source explained.
The insider added that ITV have offered to keep Castle as a "B-role", meaning that he would provide cover for Chiles, but continued: "Andrew is not sure that would be the right move for him. He might think it's time to make a change rather than being, in effect, a stand-in.
"The One Show have made enquiries, and Andrew is interested. After all, it is a hugely successful programme and it would be a new challenge for him. It would also be one in the eye for GMTV."
However, Castle's spokesperson said: "Andrew is currently under contract with GMTV. I can neither confirm nor deny that we have had approaches from anyone else."
Matt Baker and Richard Hammond have also been linked to the One Show job.
