Businesswoman Dr Bell, a former member of S4C's governing authority, said she would not be commenting on the matter.
It is understood the Welsh Government hoped another of the shortlisted candidates would be appointed in her place.
However, the Department for Culture, Media and Sport has opted to re-advertise the position.
It emerged the Welsh Government had vetoed the preferred candidate to represent Wales on the new BBC Board last week.
The members for England and Scotland were confirmed last Thursday.
But a UK government source said the Welsh Government had "seen fit to veto the secretary of state's choice of candidate".
The Welsh Government said: "Throughout the appointment process, our primary concern has been to ensure the BBC board properly represents the needs of the people of Wales.
"While we were unable to agree with the secretary of state's choice for the post of board member for Wales, we have made it clear there remain appointable candidates on whom we can agree.
"It is unfortunate that the secretary of state has been unwilling to have a proper dialogue with us about it. The latter stages of this process have been deeply unsatisfactory and we believe it is in both Wales and the BBC's interests that we now reopen the process."
Dr Bell has a background in the oil and gas industry and is a former managing director of the Global Oil and Gas Group at the Chase Manhattan Bank.
She is also a trustee of the National Museum of Wales and Wales Millennium Centre and a member of Cardiff University's council.
The candidate to represent Wales on the BBC board who was vetoed by Welsh Government was Dr Carol Bell.