The leaders of mass demonstrations have welcomed proposed talks, but still insist that Mr Leung should step down.
The protesters are angry at China's plan to vet candidates for elections in 2017, and say they want full democracy.
They have surrounded two Hong Kong government buildings, but have been warned not to try and occupy them.
At a news conference shortly before the protesters' midnight deadline for his resignation, the chief executive said attempting to occupy building would lead to "serious consequences".
He said the territory's top civil servant, Chief Secretary Carrie Lam, would open a dialogue with student leaders as soon as possible.
"Tonight, the Hong Kong Federation of Students issued an open letter asking for a meeting with the chief secretary, representing the Hong Kong government, to discuss one item - and this is the constitutional development of Hong Kong," Mr Leung told reporters.
"We have studied the letter in detail, and I'm now appointing the chief secretary to represent the Hong Kong government to meet with the representatives of the Hong Kong Federation of Students to discuss constitutional development matters."
He added: "I will not resign because I have to continue with the work for elections."
Student protest leaders responded by saying that they planned to join the talks with the government but insisted that Mr Leung should step down, saying he had "lost his integrity".
A wider pro-democracy group that has joined the demonstrations, Occupy Central, issued a statement saying it "hopes the talks can provide a turning point in the current political stalemate".
It added: "However, we reiterate our view that Chief Executive Leung Chun-ying is the one responsible for the stalemate, and that he must step down."
At the scene: Ali Moore, BBC News, Hong Kong
The late-night press conference came in the final hours of the student-imposed deadline for the resignation of CY Leung.
And while he made it clear he wasn't going anywhere, the agreement to talk has defused the immediate tensions. Protesters remain on the streets but the air of expectation in the lead-up to the deadline has gone, and the focus now is on what the government may offer when they finally sit down at the table.
The students want the talks held in public, which is an interesting challenge for the Hong Kong authorities. And while there's no indication they'll get what they want - in fact the opposite - they're sticking to their demands.
But the meeting is a chance for both sides of this political stand off to take a step back.  Whether it's enough to get the protestors to go home is another question.
Police had earlier warned protesters massed outside the Office of the Chief Executive and the Central Government Complex not to try to breach the cordon protecting the buildings.
Lord Patten, the last British governor of Hong Kong before it was handed over to China in 1997, told the BBC: "For the hundreds of thousands of demonstrators and the majority of people in Hong Kong, the big issue is whether or not they can secure the free and fair elections that they have been promised for years.
"The future of the chief executive is tied up on the outcome."
Speaking about the fear of an escalation in violence, Lord Patten added: "China has so much at stake... I can't believe this could become another Tiananmen."
Q&A: Hong Kong's democracy controversy
Hong Kong's Chief Executive CY Leung says he will not resign, but has offered talks between his government and pro-democracy protesters.