The Syrian government has said the opposition is not serious about peace.
More than 250,000 people have died in almost five years of war in Syria.
Eleven million others have fled their homes as forces loyal to President Bashar al-Assad and those opposed to his rule battle each other, as well as jihadist militants from so-called Islamic State (IS).
The opposition says its demands are in line with a recent UN Security Council resolution
“I believe we received very positive messages from the special envoy and tomorrow he will have a meeting with the regime side and we will see wait for a reply from him,” HNC spokesman Salim al-Muslat was quoted as saying by Reuters late on Monday.
Talks aimed at finding a political solution to the Syrian conflict were due to begin in earnest at the Palais des Nations in Geneva on Monday, with representatives of the warring parties sitting in separate rooms and UN officials shuttling between them.
However, Mr de Mistura’s office said his morning appointment with the government’s delegation had been rescheduled because he first wanted to have an official meeting with the HNC delegation in the Swiss city.
The envoy only paid an informal visit to the HNC negotiators’ hotel on Sunday, when he was reported to have made a proposal based on their demands.
Analysis – Lina Sinjab, BBC News, Geneva
The Syrian opposition delegation’s meeting with Staffan de Mistura marks the official start of the talks in Geneva, as the UN special envoy met the government’s representatives on Friday.
There are immense diplomatic efforts taking place behind the scenes to ensure these talks are successful, but they could easily be derailed.
The opposition High Negotiation Committee says it may pull out if the government does not stop its attacks on civilians. It cites the bombardment of the Damascus suburb of Muadhamiya on Sunday and the attack on a refugee camp in the north on Saturday as examples of regime’s unwillingness to co-operate.
Only Russia has enough leverage with the regime to deliver. It is the only world power involved in the Syrian conflict with a military base there. But it may be angered by the arrival in Switzerland of the opposition’s chief negotiator. Mohammed Alloush is a member of Jaish al-Islam, an Islamist rebel group that Russia considers a terrorist organisation.
The story of the conflict
HNC member Farah Atassi said before Monday’s meeting the group planned to give Mr de Mistura a “road map” for the implementation of their demands. The top priority should be halting the “unprecedented bombardment by the Syrian regime” of rebel-held suburbs of the capital Damascus, she said.
The HNC is also demanding the release of prisoners, in line with a UN Security Council resolution approved in December.
The head of the government’s delegation, Bashar al-Jaafari, said it was considering ceasefires and prisoner releases, but suggested they might come about as a result of the talks and not before.
The talks in Geneva are part of an 18-month peace process outlined in Security Council resolution 2254 that includes a transitional period ending with elections.
As the UN struggled to convene the talks, the fighting on the ground in Syria continued unabated.
The Syrian Observatory for Human Rights said about 60 barrel bombs had been dropped by government aircraft on the rebel-held Damascus suburb of Muadhamiya every day for the past three days.
The UK-based monitoring group also said the death toll from a suicide bombing in the government-controlled Sayyida Zeinab district had risen to more than 70.
Rebels also said they were resisting a government offensive near a supply route to the northern city of Aleppo. Troops and pro-Assad militiamen, backed by dozens of air strikes, are reported to have gained ground near the village of Bashkoy.
Meanwhile, Turkish disaster agency AFAD reported that more than 3,000 ethnic Turkmen and Arabs had crossed into Turkey over the past three days after pro-government forces advanced in the north of Latakia province.