Julie Dore, Chris Read, Sir Steve Houghton and Ros Jones will get together on Thursday aiming to thrash out an arrangement acceptable to both camps, according to Barnsley MP Dan Jarvis.
He said they were now very close to agreeing a deal that offered “the best of both worlds.”
Under the plan, Barnsley and Doncaster would give their consent to the Sheffield City Region deal, releasing money and powers, following the election of a South Yorkshire mayor on a two-year term.
At the same time, they would be free to pursue a wider Yorkshire deal if it is agreed.
Mr Jarvis has led the campaign for a ‘One Yorkshire’ ambition for devolution encompassing the county.
He said: “We are all working to negotiate the wording of an arrangement and I am hopeful that we are close to agreement.
“Regardless of what anyone thinks about the respective merits of a SCR or wider Yorkshire deal, there is a perfectly reasonable compromise to be achieved here. One which sees us get the SCR deal over the line from May 18, draws down the money and the powers, whilst retaining the flexibility for individual local authorities to pursue a wider Yorkshire deal as and when such a deal is agreed and if they choose to do so.
“This is a hugely important issue for our area and Britain’s departure from the European Union should further focus our minds on the importance of getting it exactly right. Although other parts of the country are ahead of us, it looks like we are now close to securing an arrangement that would give us the best of both worlds and allow us to move forward.”
In September, Barnsley and Doncaster pulled out of the SCR deal to pursue One Yorkshire, halting the release of £900m over 30 years and devolved powers over adult skills, employment support, housing and planning, transport, business growth and investment and infrastructure.
But pressure has been building to do a deal as government favours areas with devolution agreements including Manchester, Liverpool and the West Midlands, while Yorkshire falls behind.
South Yorkshire is on track to elect a mayor in May 3 with no money and control only of transport.
Mr Jarvis said there was still Parliamentary time to change the statute to allow a two-year mayor.
A Government spokesman said they were awaiting a formal response from the four leaders.
“We have always said we would welcome discussions on a widely-supported greater Yorkshire devolution deal provided the Sheffield City Region deal was not threatened.
“While we will not undo the Sheffield City Region deal, which has been partly implemented and would bring around £1 billion of new investment to the area, we have proposed to the four South Yorkshire leaders what we believe is a good way forward.
“We are awaiting an agreed formal response from the four leaders to the proposal that the Secretary of State set out in his letter towards the end of last year.”