A committee of MPs and Peers wants to be able to order intelligence officers to provide it with information as part of a proposed overhaul of its powers.
In its annual report, the Intelligence and Security Committee (ISC) also called on the prime minister to give up his right to see its reports first.
The committee oversees the work of MI5, MI6 and the Government Communications Headquarters (GCHQ).
It says its structure has become "significantly out of date".
The 2010/11 annual report covers issues including cyber-security, the London Olympics and the government's cost-cutting.
In it, the Intelligence and Security Committee asks that its remit be extended from covering the security agencies' policies, administration and finances to their operations as well.
Chairman Sir Malcolm Rifkind said the committee's remit and structure needed to be reformed.
Sixteen years after it was created by former Prime Minister John Major, the committee's arrangements are "significantly out of date", he says.
"Public expectation in terms of transparency and openness has increased significantly during this time, and we must ensure that the committee now has the powers and remit that are necessary to provide reassurance to the public and to Parliament," he said.
"It is essential that we are able to provide credible reassurance that, consistent with necessary secrecy and security, the agencies operate in the public interest."
The ISC report says that it is no longer appropriate that the committee be hosted by the Cabinet Office.
"As a matter of principle, no matter what the circumstances, it clearly is not right to be hosted by an organisation that you have some role in overseeing and there is a danger that boundaries might not be respected."
Sir Malcolm says the ISC should become a parliamentary committee, delivering its reports to Westminster as well as Prime Minister David Cameron.
He says the proposed reforms will "increase accountability, transparency and capacity for oversight of the intelligence community".
The changes could be introduced in the government's Green Paper on the protection of intelligence material in the courts, he says.