Sara Khan, who has campaigned for women's rights in Muslim communities, has been given the task of rooting out extremism in the UK.
She has promised "zero tolerance to those who promote hate".
But her support for the Home Office's Prevent strategy has led to claims she is too close to the government.
Downing Street has insisted Ms Khan is "expertly qualified" for the new role - but a number of Muslim organisations are understood to be calling for her to be sacked and saying they will not work with her.
A petition by anti-Islamophobia campaign MEND, saying her appointment "will further damage relations between the government and Muslim communities," is being circulated, the BBC understands.
Lady Warsi, the first Muslim woman to serve as a British cabinet minister, said many British Muslims saw Ms Khan as a "mouthpiece" of ministers.
The Conservative peer questioned Ms Khan's likely independence as the Commissioner for Countering Extremism in a series of tweets and warned of "destructive and dangerous games" being played. [BBC] Read more
