"A very unscientific straw poll just as people were emerging from the polling booth would suggest that most tend to be voting against this proposal," he said.
"The last opinion poll said around 52 [or] 53 per cent of the population were against the proposal, although the People's Party still insist that this is going to be a very close vote."
Supporters of the proposed ban say minarets represent the growth of an alien ideology and legal system that have no place in the Swiss democracy.
"Forced marriages and other things like cemeteries separating the pure and impure - we don't have that in Switzerland, and we do not want to introduce it," Ulrich Schlueer, co-president of the Initiative Committee to ban minarets, said.
"Therefore, there's no room for minarets in Switzerland."
But Switzerland's Muslims have said that the referendum is fuelling anti-Islamic feeling in the country.
"The initiators have achieved something everyone wanted to prevent, and that is to influence and change the relations to Muslims and their social integration in a negative way," Taner Hatipoglu, the president of the Federation of Islamic Organisations in Zurich, said.
"We are frightened, and if the atmosphere continues to be like this and if the anti-Islamic hate increases, then the Muslims indeed will not feel safe anymore. This of course is very unpleasant."
About 400,000 Muslims live in Switzerland - whose population is just under eight million. Most Muslim citizens are immigrants from the former Yugoslavia and Turkey.
Although Islam is the country's second largest religion after Christianity, there are only four mosques with minarets in the whole country.
'Know your place'
Posters have appeared in many Swiss cities showing a dark, almost menacing figure of a woman, shrouded from head to foot in a black burka. Behind her is the Swiss flag, shaped like a map of the country, with black minarets shooting up out of it like missiles.
The cities of Basel, Lausanne and Fribourg banned the billboards, saying they painted a "racist, disrespectful and dangerous image" of Islam.
The United Nations Human Rights Committee called the posters discriminatory and said Switzerland would violate international law if it bans minarets.
Our correspondent said that there was a political message behind Sunday's referendum.
"The reality is, as was described to me by a Swiss resident who is not a Swiss citizen, this is the right-wing Swiss People's Party sending out a message to Muslims: 'Know your place in Switzerland'," he said.
"They believe, the right-wing People's Party, that if the Muslims get their mosques and their minarets it will follow on that they will want, perhaps, separate schooling and there could be a campaign to turn Switzerland, of all places, into a place that practices Sharia."
The Swiss government and business leaders have opposed a minaret ban, saying it would be harmful to the country's image abroad and disastrous for the Swiss economy.
The Swiss People's Party forced the referendum after collecting 100,000 signatures within 18 months from eligible voters supporting the motion.