South Yorkshire councils signed a devolution deal with then chancellor George Osborne last year which should see a mayor for the area elected in 2017 in exchange for new powers and money but that could now be reviewed.
Downing Street attempted to play down reports that the new prime minister is ready to scrap the commitment to mayors but Yorkshire council sources confirmed the issue is being discussed by Whitehall officials.
South Yorkshire council figures expressed concern that backtracking on mayors could be accompanied by a wholesale re-writing or watering down of the devolution deals struck by Mr Osborne.
A Sheffield City Council spokesman said: “This Government made it clear that city regions needed to have an elected mayor if they were to be persuaded to back economic growth in cities like Sheffield, and if the same Government now chooses to reverse this policy then so be it, but they need to decide which way they are facing, and be consistent.
“If they do back off now then they must still back our plans for growth and we will relentlessly hold them to account for that.”
Mr Osborne and David Cameron always insisted they were not imposing elected mayors on areas but council leaders were left in no doubt that the measure was expected in return for a meaningful devolution deal.
The signal that Mrs May may be prepared to review the issue was welcomed in the rest of Yorkshirewhere devolution talks have been deadlocked.
North, West and East Yorkshire councils have struggled to agree whether there should be a single devolution agreement for the area or a series covering smaller areas.
It is thought removing the obligation to have an elected mayor could reduce some of the political tensions which have made reaching agreement more difficult.
North Yorkshire County Council leader Carl Les said: “Any flexibility in the process would be helpful if it means leaders can get round the table and discuss it in more detail.
“I remain convinced that the larger Yorkshire footprint, whatever it is called, is the right approach to devolution to Yorkshire.”
Hull City Council leader Stephen Brady has come under fire after revealing he has not spoken to West Yorkshire counterparts for months over devolutuon plans.
Liberal Democrat group leader Mike Ross said: “It is all very well for coun Brady to say he hasn’t spoken to the West Yorkshire leaders for a year, but he needs to demonstrate what he has done to speak to them.
“We have been pushing for the council to get sorted on a deal for the area or there is a real danger of Hull getting left behind.
“Hull could lose out massively on future funding, all because Labour thought they were too big to deal with our neighbours properly and at the same time have failed to make any decent case to West Yorkshire.
“Sadly, whenever I speak to anyone in the Yorkshire region about it, too often the problem is a reluctance to work with Labour run Hull.”