UK intelligence agencies should be allowed to retain controversial intrusive powers to gather bulk communications data but ministers should be stripped of their powers to authorise surveillance warrants, according to a major report on British data law.

The 373-page report published on Thursday – A Question of Trust, by David Anderson QC – calls for government to adopt "a clean-slate" approach in legislating later this year on surveillance and interception by GCHQ and other intelligence agencies.

However, Downing Street hinted that David Cameron was unlikely to accept one of his key recommendations: shifting the power to agree to warrants from home and foreign secretaries to a proposed new judicial commissioner.

The prime minister's spokeswoman said the authorities needed to be able "to respond quickly and effectively to threats of national security or serious crime", which appears to suggest ministers are better positioned to do this than judges.

Anderson's report, commissioned by Cameron last year, comes in response to revelations by the US whistleblower Edward Snowden about the scale of government surveillance disclosed two years ago.

Anderson, introducing his report, said: "Modern communications networks can be used by the unscrupulous for purposes ranging from cyber-attack, terrorism and espionage to fraud, kidnap and child sexual exploitation. A successful response to these threats depends on entrusting public bodies with the powers they need to identify and follow suspects in a borderless online world.

"But trust requires verification. Each intrusive power must be shown to be necessary, clearly spelled out in law, limited in accordance with human rights standards and subject to demanding and visible safeguards".