Earlier this month, Sir Keir committed Labour to reinstating housebuilding targets that were scrapped by Prime Minister Rishi Sunak after a backbench revolt.
Ministers backed down and made the target of building 300,000 houses a year in England advisory rather than mandatory, taking the pressure off local authorities to approve new developments.
Sir Keir told the newspaper that, in doing so, the Conservatives had “killed the dream” of homeowning for “a whole generation”.
“It will fall to us to deal with that.”
Green belt policy aims to prevent urban sprawl by keeping land permanently open.
There are currently 13 green belts in London and 1 in Wales, all of which are protected by planning controls as well as an additional presumption against “inappropriate development” within their boundaries.
Sir Keir said Labour would “fix what is essentially a broken system” giving councils and residents the power to build on certain areas of green belt land if they wish.
He added: “It cannot be reduced to a simple discussion of will you or will you not build on the green belt. This is why it’s important for local areas to have the power to decide where housing is going to be.
“Very often the objections that people have to housebuilding on the green belt are valid because the control by landowners and developers mean that the houses are proposed in areas where it’s quite obvious that there’s going to be a local concern.
“Give local authorities, local areas more power to decide where it will be and you alleviate that problem. So it’s not as binary or straightforward as ‘green belt, not green belt’. It’s how you direct where the housing will be.”
During the Tory leadership contest last summer, Mr Sunak said the green belt was “extremely previous” and said his planning policy would be “brownfield, brownfield, brownfield”.
It comes hours after Housing Secretary Michael Gove conceded there “simply aren’t enough homes” in Britain.
Across Britain, the average asking price for a first-time buyer type property currently sits at a record of £224,963, according to data shared earlier this month by property website Rightmove.
Cabinet minister Mr Gove said: “There is a problem and the problem is there simply aren’t enough homes in this country.
“It is increasingly difficult to get on the property ladder.”