The announcement came after Mr Straw became the first senior Labour politician to say he was willing to appear on the show with Mr Griffin.
The BBC then confirmed that the two men were among the panellists booked for a recording of the show, hosted by David Dimbleby, in London on October 22.
Controversy was sparked earlier this month when the BBC announced it would be willing to feature representatives of the BNP on Question Time after the party won two seats in the European Parliament in elections in June.
Labour reviewed its approach of refusing to share a platform with the far-right BNP and Gordon Brown made clear he was ready to allow a minister to take on Mr Griffin, now an MEP for North West of England. But cabinet ministers such as Peter Hain and Alan Johnson said they would not go on Question Time if the BNP leader was invited.
Yesterday, Mr Straw said: “Wherever we have had BNP problems in my area and when we have fought them hard, we’ve won the seats back. That’s what we have to do. We’ve got to make the argument for people and I am delighted to do so.”
Tony Kearns, assistant general secretary of the Communication Workers’ Union, said it was a “disgrace” the BBC was going ahead with offering the BNP a seat on Question Time. He called on government ministers and MPs to protest against the decision.
Peter Hain, the Welsh Secretary, is reported to have said he will be meeting BBC executives on Wednesday to try to persuade them to reverse their decision. He also questioned whether the BBC was legally obliged to invite the BNP to appear.
“I don’t buy this stuff that under electoral law the BNP’s two European seats allows them to be treated in the same way as the mainstream parties” he said. “It’s an absolute scandal the BBC are treating an avowedly racist and fascist [party] on equal terms, in a way that will legitimise their standing in the public’s eyes.”
Mr Hain said he was directing his anger at the BBC, rather than Mr Straw, saying it had put Labour ministers in an “impossible position” because the party did not want the BNP to go unchallenged.
Unite Against Fascism has pledged to hold a demonstration against the BBC’s decision, with Mr Hain adding he fully supported the group’s actions. A BBC spokeswoman said its obligation to treat all parties registered with the Electoral Commission “with due impartiality” was enshrined in the corporation’s charter.
Other members of the Question Time panel have not been confirmed, but the LibDems said they will probably field Chris Huhne, their home affairs spokesman.
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