It comes as Ms Aitken raised concerns Scotland’s largest city had been sidelined by the Chancellor’s spending review last month.
The Treasury confirmed a £160 million Investment Zone in the Glasgow Region and £20 million for Trailblazer Communities, however the city council leader warned it fell well short of the funding handed to English regions.
Mr Murray acknowledged the concerns but said June’s spending review issued £52 billion for Scotland, which was “more money than ever before” to enable the Scottish Government to invest in public services.
However, Finance Secretary Shona Robison argued the country had been “short-changed” by the funding announcements.
Mr Murray’s letter, seen exclusively by The Herald, said: “Devolution within Scotland is a matter for the Scottish Government – as are many of the relevant policy areas such as housing, skills and transport – but we would be delighted to work with them to help ensure the Glasgow City Region has the tools you need to deliver change and unlock the same levels of growth as your English counterparts in Greater Manchester.
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“We are always open to constructive discussions with the Scottish Government to reverse its centralisation policy and devolve powers to our cities and regions.
“I would welcome a joint letter between ourselves to the First Minister to kick start this conversation.”
The Scottish Secretary offered his team to draft a letter to the First Minister if Ms Aitken agreed.
But last month, Ms Aitken accused the Chancellor of of taking a “retrograde step” for devolution in Scotland that risked “disempowering” Glasgow.
Her concern related to detailed commitments outlined in the spending review which expanded integrated settlements for English city regions.
It meant city regions in England would not have to apply for individual grants through competitive bidding processes.
Instead, the designated cities are set to receive long-term funding that allows Mayors greater autonomy in making their own investment decisions.
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Following the spending review, these settlements are being expanded to include London, the North East, West Yorkshire, South Yorkshire, and Liverpool City Region.
Almost 40% of England’s population will now have local control over this funding, with these regions joining existing arrangements in Greater Manchester and West Midlands.
Glasgow, meanwhile, was the only Scottish region selected to be part of the National Wealth Fund which will help local leaders develop investment and growth priorities.
However, the city also has to bid for funding for Westminster alongside other parts of the UK.
In a letter seen by The Herald last month, Ms Aitken criticised the spending review.
She said: “It is clear from the Spending Review that the UK Government recognises the best way to support economic growth of English City Regions is through an integrated settlement, allowing places the ability to make their own investment decisions.
“And yet Glasgow City Region, which is larger in population, size of economy, opportunity and need than most of the Mayoral Combined Authorities, is reduced to simply administering programmes on behalf of UK Government as if it were a small local authority.”
She added: “The empowerment of our comparator city regions in England and the disempowerment of Glasgow City Region threatens all of the progress we have made. We have a shared priority of growing Scotland’s economy and ensuring our people reap the benefits of that.
“We cannot grow Scotland’s economy without growing Glasgow’s economy — and yet yesterday’s budget will not contribute to that growth and will cause us to fall behind our English counterparts.”
Finance and Local Government Secretary Shona Robison said: “The UK Spending Review document sets out in black and white that our funding for day-to-day spending is set to grow by only 0.8% over the next three years, compared with 1.2% average growth for UK Government departments. This will short-change us by £1.1 billion pounds which could make a real difference to our communities and councils, so the Secretary of State should and could be urging the Chancellor to reverse that if he wants to help Scotland.
“Councils play a crucial role in our communities which is why we jointly launched the Local Governance Review with COSLA to ensure communities have greater control and influence over decisions that affect them most.
“We are also committed to working with local authorities to deliver greater regional empowerment on decision making and investment and are working with partners to explore ways of devolving further powers to Regional Economic Partnerships, including Glasgow City Region.
“In addition, the Scottish Government has delivered a wide range of fiscal powers for local councils including greater powers within planning, parking charges, workplace parking and in Council Tax, being able to charge up to 100% on second and long term empty homes.”
Finance Secretary Shona Robison said: “The UK Spending Review document sets out in black and white that our funding for day-to-day spending is set to grow by only 0.8% over the next three years, compared with 1.2% average growth for UK Government departments. This will short-change us by £1.1 billion pounds which could make a real difference to our communities and councils, so the secretary of State should and could be urging the Chancellor to reverse that if he wants to help Scotland.
“Councils play a crucial role in our communities which is why we jointly launched the Local Governance Review with COSLA to ensure communities have greater control and influence over decisions that affect them most.
“We are also committed to working with local authorities to deliver greater regional empowerment on decision making and investment and are working with partners to explore ways of devolving further powers to Regional Economic Partnerships, including Glasgow City Region.
“In addition, the Scottish Government has delivered a wide range of fiscal powers for local councils including greater powers within planning, parking charges, workplace parking and in Council Tax, being able to charge up to 100% on second and long term empty homes.”