Saudi women are holding a day of action to challenge the kingdom's ban on female driving, amid signs of slowly growing readiness by the authorities to consider reform in the face of strong opposition by the clerical establishment.

Twitter, Facebook and other social media have been used to get women drivers on the roads on Saturday in a marathon push against this unique restriction.

Activists say they have 16,600 signatures on an online petition calling for change. Efforts to publicise the issue by the "October 26 driving for women" group have been described as the best-organised social campaign ever seen in Saudi Arabia, where Twitter has millions of users and is used to circulate information about the monarchy and official corruption.

Now the mainstream press is getting involved too, a telling indication of a thaw on this issue. "It's time to end this absurd debate about women driving," wrote Dr Thuraya al-Arid in al-Jazirah newspaper. In another paper, al-Sharq al-Awsat, Mshari al-Zaydi said: "The time has come to turn the page on the past and discuss this issue openly."

Previous attempts to promote change fizzled out in arrests for public order offences and demoralisation. In 2011, activist Manal al-Sharif made a YouTube video urging women to drive their own cars, and was imprisoned for over a week. But the signs are far more positive now.

Three female members of the shura (advisory) council – among 30 appointed by the 90-year-old King Abdullah – recommended this month that the ban be rescinded, though no debate has yet taken place. Latifa al-Shaalan, Haya al-Mani and Muna al-Mashit urged the council to "recognise the rights of women to drive a car in accordance with the principles of sharia [Islamic law] and traffic laws".

The three – praised by supporters for "stirring the stagnant water" – framed their argument with careful references to fatwas (religious edicts) banning women from being in the company of an unrelated male (a driver). Other ideas designed to reassure critics are appointing female traffic police and driving instructors. Cost is another big factor, with families having to employ chauffeurs, as is convenience.

Writer Maha al-Aqeel, who is planning to take her Mazda out for a spin in Jeddah — with her brother or nephew — sees the issue as the thin end of a wedge of reform in Saudi Arabia. "Driving is such a visible and symbolic thing," she told the Guardian. "It's not like women on the shura council – you cannot see that and you cannot see advances for women in the workplace. Many conservatives feel that if women get the right to drive then that's it, the last bastion of male control will fall. I think it should lead to other changes. That's why those who oppose it are so vehement. And that's why the government is treading so carefully. It does not want to cause a big uproar."

Signs of powerful opposition are easy to detect. This week 150 clerics and religious scholars held a rare public protest outside the king's palace in Jeddah to object to "westernisation" and "the conspiracy of women driving", blaming the US – a byword in traditionalist circles for anything distasteful or immoral – for being behind the campaign.

Until then, the government had conspicuously refrained from cracking down. But on Wednesday the interior ministry issued a stern pre-emptive warning that unlawful assemblies and marches would not be tolerated, and invoked the danger of sedition. On a closer reading, activists noted, the ministry did not actually attack the idea of dropping the ban. "They are not saying clearly that women shouldn't drive," Aqeel said. "The government wants to stay on the middle ground."

Neither sharia law nor national traffic regulations explicitly ban women from driving, but they are not issued licenses.

Campaigners have been emboldened by the low-key official response, with some emulating Sharif and uploading films on social media of themselves driving. In a video posted by the blogger Eman al -Nafjan, a female driver is seen cruising down a relatively busy road while passing motorists give enthusiastic thumbs-up signs in support. As expectations mount, many Saudi fathers are teaching their daughters to drive. "People are positive that things are going to change," said the journalist Abeer al-Mishkhas. "They just hope it will come soon. The government says it is waiting to see if society is becoming more tolerant."

Arguments aimed at keeping women off the roads can be shocking and nonsensical. "If a woman drives a car that could have negative physiological impacts as...physiological medical studies show that it automatically affects the ovaries and pushes the pelvis upwards," warned Sheikh Saleh bin Saad al-Lohaidan.

Nafjan argued in a commentary for Amnesty International that the fundamental issue was challenging patriarchy. "If there was one word to describe what it is like to be a Saudi woman, it would be the word patronising. No matter how long you live, you remain a minor in the eyes of the government."