Acting Secretary General Mirza Fakhrul Islam Alamgir made the observation hours after Sheikh Hasina's media call on Friday over her New York visit.
Hasina said Siddique will have to leave the Cabinet for his anti-Hajj remarks.
She said some procedures must be followed in removing a minister but the removal would have to wait since the president was performing Hajj.
Fakhrul said, "The prime minister spoke of taking actions as per Rules of Business. This makes clear that she cunningly skirted the Latif Siddique issue."
At a programme in New York on Sept 28, Siddique said he was "firmly opposed" to the Hajj, which he said was a 'waste of money', Tablig and the Jamaat-e-Islami.
The comments were not received well and several parties, including the BNP, demanded his prosecution.
Several cases have been filed against the telecoms minister in Bangladesh for 'hurting religious sentiments'.
"One gets seven-year jail term for defamatory comment on the prime minister but no action has been taken so far for making derogatory remark on the Prophet," said Fakhrul.
He said the government could not avoid responsibilities for the Awami League leader's comment.
The BNP spokesperson renewed their demand for arresting and prosecuting Siddique under existing law without further delay.
Asked how the minister could be arrested from abroad, Fakhrul said the government can take help from the Interpol.
"The nation has been frustrated and angered at the prime minister's speech," he said. "We did not find the slightest hint of effective steps being taken [against Siddique] in it."