"So we have taken a decision to move towards the presidency where a protest will continue on a round-the-clock basis," he said.
Protesters intend "to continue a permanent rally outside the presidential palace and the parliament", Helena Bedwell, a journalist in Tbilisi, told Al Jazeera.
"They decided to put up some tents here [outside the presidential offices] meaning they want to stay overnight. They also plan to put up so-called 'prison cages' to symbolise imprisoned democracy."
Demonstrators are angry at Saakashvili's handling of last year's Russia-Georgia war and accuse him of failing to run a democratic government.
Call for dialogue
The protests began in Tbilisi last Thursday when 60,000 people took to the streets and rallied outside parliament, waving flags and chanting "resign".
But the numbers of demonstrators dropped away over the weekend, with only 4,000 gathering on Saturday.
Saakashvili has vowed to serve out his second term, which ends in 2013.
He has urged opposition leaders to enter into talks with the government.
Opposition leaders are unhappy with Saakashvili following last year's conflict with Russia, in which Georgia lost territory as separatists and their Russian allies took full control of the breakaway regions of South Ossetia and Abkhazia.
Critics have also accused him of not implementing political changes that he promised during a revolution in 2003, in which Saakashvili came to power.
They say he has monopolised power and exerted pressure on the judiciary and the media.
In November 2007, riot police used rubber bullets, tear gas and water cannon to disperse thousands of anti-government protesters, a move that prompted condemnation of Saakashvili's actions.