They will consider plans by Third Energy to frack a well at a site near the village of Kirby Misperton in the district of Ryedale.
Campaigners say approval would set a “dangerous precedent”.
But the company says its operations are no risk to people or the environment.
Fracking involves injecting water, sand and chemicals at high pressure into rocks deep underground to unlock trapped gas.
The process was banned in 2011 after the shale gas firm Cuadrilla caused minor earth tremors when it fracked a well in Lancashire.
Third Energy has licences to produce gas in North Yorkshire and offshore in the North Sea.
In 2013 it drilled an exploratory well near the village of Kirby Misperton, close to the North York Moors National Park.
It now wants to frack the well to test whether it can unlock shale gas from rocks up to 10,000ft (3,000m) underground.
‘All we want to do is test whether gas will flow from an existing well using hydraulic fracturing,’ says Rasik Valand, chief executive of Third Energy.
‘What we are planning is safe. There is no risk to the environment or the public.’
But there has been strong opposition to the plan from the local community and environmentalists.
Hundreds of anti-fracking protesters are expected to gather outside the council meeting in Northallerton on Friday.
The council has received more than 4,000 representations, the vast majority against the proposal.
Green campaigners fear that approval here would open the door for fracking elsewhere.
‘It risks setting a dangerous precedent for Britain,’ says Craig Bennett, chief executive of Friends of the Earth.
‘We strongly urge councillors to reject this planning application which is clearly deeply unpopular with the local community.’
Others are concerned about the impact on the local economy as Kirby Misperton is also home to Flamingo Land, one of Yorkshire’s top tourists attractions.
‘This place is beautiful,’ says retired bishop, the Right Reverend Graham Cray, who lives in the village. ‘The economy is based on tourism and agriculture. Tourists don’t come to a gas field and that is what we will turn into if we get the scale of fracking that is intended.’
Last year Lancashire County Council rejected Cuadrilla’s plan to carry out exploratory drilling and fracking at two new sites on the Fylde.
A planning inquiry into that application took place earlier this year. A recommendation will be sent in the summer to the communities secretary for a final decision.
The Third Energy application involves an existing site that has produced gas since the 1990s.
The well has already been drilled.
The shale gas produced will also be sent through a network of pipes to a nearby power station. That will avoid air pollution caused by flaring gas.
So the company expects the fracking activity to be completed within eight weeks.
The council’s planning officer has recommended that the application be approved. But it is now down to the 11 councillors on the planning committee to take the final decision.
That decision may not come until Monday due to the number of people who have registered to speak at the meeting.
If approved, fracking could start by the end of the year.
The Department of Energy and Climate Change refused to comment on Third Energy’s application ahead of the council meeting.
But in a statement it said: ‘Shale gas is a fantastic opportunity which will create jobs, boost our economy and strengthen our energy security.
‘There is tough regulation in place to ensure that fracking can take place safely – now is the time to start exploration and find out just how much shale gas is there and how much we can get out of the ground.’