Protesters were demanding Lam scrap the extradition bill and authorities apologize for the police actions.
"I personally have to shoulder much of the responsibility".
The Beijing-controlled newspaper Ta Kung Pao said on Tuesday morning that Lam would hold a press conference to apologise again and formally announce the law's withdrawal, but hours later withdrew the article without providing further detail.
Protesters also want her to release all of the arrested demonstrators, to cease referring to the protest as a "riot" and to investigate police for excessive violence against protesters, which has included the use of tear gas.
A key protest leader who is a Christian, Joshua Wong, says China's extradition plan has blown up in its face, reigniting the protest movement against dictatorial rule from China.
Hong Kong's Chief Executive Carrie Lam Cheng Yuet-ngor said she will redouble her efforts in serving the special administrative region and improving livelihoods and governance in the city for the remainder of her term.
"In recognition of the anxiety and fears caused by the bill in the last few months, if we don't have confidence from the people we will not proceed with the legislative exercise again", Ms Lam said.
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"If the bill does not make legislative council by July next year, it will expire and the government will accept that reality", she promised.
Protest organisers were unmoved by Lam's latest public statement, and slammed her for failing to address their demands.
"Carrie Lam is continuing to lie", said Jimmy Sham, with the Civil Human Rights Front, Reuters reported.
Critics say the legislation would expose people in Hong Kong to China's deeply flawed justice system and lead to further erosion of the city's judicial independence.
Lam's apology was immediately and ostentatiously accepted by her political supporters and pro-Beijing politicians, and just as quickly dismissed by protest leaders, who denounced her as insincere, duplicitous, and too badly damaged to continue as chief executive. "He won't give in easily", said Beijing-based political analyst Hua Po, noting that the extradition bill was merely suspended, not dropped. It started when she ignored opposition from Hong Kong's business community shortly after the ordinance was proposed in February - prompting many wealthy individuals to move capital to safe havens such as Singapore.
But many residents fear that they are being slowly eroded by what they feel is Beijing's tightening grip on the semi-autonomous city.
As recent evidence, they offer the failure of the huge pro-democracy "Umbrella Movement" in 2014, the imprisonment of protest leaders and disqualification of popular lawmakers, and the disappearance of booksellers critical of Beijing.
Some have characterized the conflict between China and Hong Kong as a conflict between communism and Christianity, and an unofficial anthem of the protests was "Sing hallelujah to the Lord".
Asked by CBS News during her news conference whether her decision not to retract the bill might deepen distrust in her among Hong Kong residents, Lam said she would "not proceed again with this legislative exercise if these fears and anxieties can not be adequately addressed".