Mr Prescott said: "He will enter into discussion with the bidders with our full co-operation.
Both sides are claiming success after talks over the funding of the Tube's modernisation produced a breakthrough.
"I have got what I want which is the safety of Londoners secured by retaining Bob Kiley as the management controller at all levels."
Modernisation plans in train
The controversial plans to part-privatise the Underground seem to be going ahead, but with the crucial difference that the firms bidding for contracts to modernise stations and track will now not be involved in the management of the Tube network.
But the implementation of the plans will be delayed for several weeks as a result of calls for more detailed talks.
Deputy Prime Minister, John Prescott promised that as part of the deal Mr Kiley will now receive all the commercial documents involved, enabling him to make a reasoned judgement on the issue. Some papers had previously been withheld.
Mr Prescott said: "He will enter into discussion with the bidders with our full co-operation.
"We do not envisage this going into months of delay. It is at the first stage now and it will not take very long to decide if he has a solution for London and for Ken Livingstone."
Strike action
The news comes as London faces the first of three Tube strikes on Monday, with unions protesting over the issue of safety which they say a partial privatisation could jeopardise.
But Mr Prescott insisted: "We believe that PPP (Public Private Partnership) is the best way to deal with this but we now want to subject that to the examination of Bob Kiley and London Transport."
The gap between the two sides on the issue of privatisation had previously been huge, with Mr Kiley calling John Prescott's earlier plans "fatally flawed".
Now Mr Kiley says the two sides have moved much closer together, although he admits "gaps" still remain.
Breakthrough
Heralding what he said was the "big breakthrough," Mr Livingstone said: "For me the most important thing is the unified management structure.
"The question of whether we have bonds is secondary to that.
"The thing that was not negotiable was splitting up the Underground we now have that secured."
Outlining the crucial role to be played by his transport commissioner in the future the mayor added: "Bob Kiley's agreement will be necessary for the next stage and he is not going to buy a deal which would endanger Londoners."
Speaking for the Conservatives Bernard Jenkin, the shadow transport minister said: "After four wasted years the government has been forced to admit that there are fundamental flaws in their undeliverable PPP."
He then predicted more delays to modernisation saying: "There will be no new investment in the Tube until after the general election."
Talks
The announcement follows over two and half hours of talks between the deputy prime minister and Mr Kiley on Thursday.
Plans for a sell-off have come under fire from numerous sources, including Mr Livingstone and the London assembly, a committee of MPs and the National Audit Office.
Polls of Londoners also suggest the public has little taste for a sell-off.