Shadow home secretary Yvette Cooper claimed the security services' hands were tied for nearly five years by Mrs May's "wrong" decision to scrap powers to move terror suspects away from their extremist networks.
Relocation powers have been reintroduced by the coalition this year but Ms Cooper called for the security services to immediately brief the Intelligence and Security Committee on how the loss of the measures might have affected their work.
This would allow MPs to review whether the loss of relocation powers had led to more British jihadists travelling to Syria and Iraq and more radicalisation in the UK.
Ms May told BBC1's Andrew Marr Show:
Labour's shadow home secretary Yvette Cooper believes the rape tweets directed at Jessica Ennis-Hill prove why her stance on Ched Evans is so important.
Ennis-Hill received abusive tweets from trolls on Friday after making it clear she would remove her name from a stand at Sheffield United if the club offer the convicted rapist a contract.
The tweets directed at the Sheffield born Olympic Champion are currently being investigated by South Yorkshire police.
Cooper also urged the League One club to "recognise their responsibility to the local community" by resisting the temptation to offer Evans a professional deal.
Shadow home secretary Yvette Cooper has said Labour would "look carefully" at any anti-terror proposals the Prime Minister brings forward.
"The Government should be introducing mandatory de-radicalisation programmes as a priority," Yvette Cooper said.
"Our border controls are simply not picking people up, and no new orders will change that. Only this week a suspect on bail managed to flee with his passport, reportedly to Syria, Cooper said."
“The Prime Minister would be well-advised to reinstate measures we know work, and we hope some of these changes will be in any proposed legislation.”
Labour's shadow home secretary Yvette Cooper has backed Jessica Ennis-Hill's promise to remove her name from a stand at Sheffield United's Brammall Lane stadium should the club re-sign the convicted rapist Ched Evans.
Labour would reverse cuts to the police planned for next year if it wins power in May, shadow home secretary Yvette Cooper has said.
The party would also scrap police and crime commissioners (PCC) to pay to protect women's refuges.
Shadow Home Secretary Yvette Cooper has said that South Yorkshire Police Commissioner Shaun Wright ought to stand down, but stressed that reforms were needed too:
She also called for the "child abuse inquiry to start urgently".
Rape cases that are dropped by the police would be reviewed under Labour plans to boost the rights of victims, Yvette Cooper will say.
The shadow home secretary will pledge to give victims a "strong voice" and warn the low rate of allegations being pursued is "shameful".
In a speech to the Fabian Society, she will insist Labour is focusing on reforms that give more power to individuals and communities but stand ready to make tough decisions about making savings.
"Just as in the criminal justice system, across our public services, ministers are driving fragmentation, reorganisation and competition just at a time when local services need to collaborate more to deliver better services and save money too", she is expected to say.
Parents, schools, sports clubs and other organisations caring for young children need to have "confidence in the vetting and barring system" so they know employees pose no threat to youngsters, the Shadow Home Secretary has said.
Yvette Cooper hit out at Home Secretary Theresa May who she claimed was "warned repeatedly" that changes she made to vetting and barring laws "left major loopholes in the system".
Shadow home secretary Yvette Cooper will warn the Government it must not "bury its head in the sand" as she calls for reforms to keep up with ever-evolving cyber-crime.
In a speech to the Demos think tank, Ms Cooper will call for a new national strategy for tackling online fraud, tougher action to tackle online child pornography and an overhaul of parliament's Intelligence and Security Committee, which keeps a check on the work of the intelligence agencies.
"The oversight and legal frameworks are now out of date. That means we need major reforms to oversight and a thorough review of the legal framework to keep up with changing technology," she is expected to say.
"Above all we need the Government to engage in a serious public debate about these new challenges and the reforms that are needed."