The director general of the BBC, Tony Hall, has rejected demands from a cross-party group of MPs, including Boris Johnson and Alex Salmond, to stop the broadcaster using the term “Islamic State” to refer to the terrorist group.

The MPs made their demand in a letter following criticism of the BBC from David Cameron, who used an appearance on the Today programme on Monday to suggest that Muslim listeners would “recoil every time they hear the words Islamic State” to refer to its “appalling, barbarous regime”.

Initiated by Rehman Chishti, the Conservative MP for Gillingham and Rainham, the letter urged the BBC to instead adopt the term “Daesh”, based on Arabic acronym al-Dawla al-Islamiya fil Iraq wa’al Sham, which translates as Islamic State of Iraq and Sham (Syria), but is close to “Dahes” or “one who sows discord”. It was signed by 120 MPs including the London mayor Boris Johnson and chair of the home affairs select committee Keith Vaz.

But the head of the BBC rejected the demands, saying that using Daesh would not preserve the BBC’s impartiality as it risked giving an impression of support for the group’s opponents, the Times reports. He is said to claim that the term is used pejoratively by its enemies.

Instead, it is reported, Hall said the BBC would use terms such as the “Islamic State group” to distinguish it from a true state, and continue to use descriptions such as extremist or militant for its members.

Chishti told the Times that Hall’s decision not to adopt the term Daesh was very surprising and rather unacceptable, as it has been adopted across the Middle East and by the French and Turkish governments.

In a Commons debate later on Monday on the UK’s response to the Tunisia massacre, the prime minister said it would even be preferable for the broadcaster to use the term Isil, which is used by most UK politicians. Isil is short for Islamic State in Iraq and the Levant, the description used by British security agency MI5.

“I personally think that using the term Isil or ‘so-called’ would be better than what the BBC currently does,” Cameron said. “I don’t think we’ll move them all the way to Daesh, so I think saying Isil is probably better than Islamic State because it is neither, in my view, Islamic or a state.”