The Localism Bill aims to deliver a legacy of empowered cities that can drive private sector growth and jobs.
The changes means city leaders, alongside the area's local enterprise partnership, can make the case for being given new powers to promote economic growth and set their own distinctive policies.
This amendment opens the door to greater local control over investment to drive growth, for example for housing and planning, economic development, or pooling resources and effort across functioning economic areas.
It means cities can be more joined-up about local investment, moving on from case by case funding applications, saving time and money.
Ministers are listening to what cities say they need as the Government seeks to rebalance the economy, by empowering local civic and business leaders to create jobs and growth.
Instead of imposing solutions the Government will strike deals with cities that give local authorities freedoms to set their own policies where they make a strong case for greater devolution.
According to independent forecasts, if the Core Cities were given greater freedoms they would be able to deliver an extra one million jobs and £44billion to the economy in the next decade across their local enterprise partnership areas.
The UK's 'Core Cities' urban areas - Bristol, Birmingham, Leeds, Liverpool, Manchester, Newcastle, Nottingham and Sheffield - are the main drivers of the economy outside London and the South East, delivering 27 per cent of the national economy.
Greg Clark said:
"Dynamic cities are essential to faster economic growth across the UK so it's vital they have the means to exploit their potential.
"We are determined to see UK cities given greater control over their economic destiny, allowing them to compete on the global stage.
"In the future UK cities will be able to bid for freedom to set their own distinctive policies. When they come up with innovative proposals for doing things differently, we will devolve the powers cities need to drive private sector growth and create jobs."
Chris Murray, Director of the Core Cities Group, added:
"Core Cities very much welcome the Government's acceptance of this amendment, which has achieved good cross-party support. The performance of our biggest cities is critical to the national economy, and we know that there is a strong relationship between the levels of decentralisation cities have in other countries and how competitive they are.
"At a challenging economic moment, we need to set our cities free to really deliver, and this change in the law will allow that to happen."
