Opposition leaders said protesters would stay in front of the residence in central Tbilisi day and night until Saakashvili, criticized for last August's war with Russia and authoritarian trends, stepped down.
"We came to the president, who is not hearing us well," opposition Conservative Party leader Kakha Kukava said.
Protesters and some opposition leaders have stayed near the parliament building.
The campaign began on Thursday, when tens of thousands of people took to the street. On Friday, people blocked Tbilisi's main roads running past the president's residence and the public broadcaster's headquarters as part of a campaign of civil disobedience.
Security at the presidential residence has remained heightened, with masked riot police and water cannons deployed in front of the building.
The opposition also said on Monday that it would expand its campaign by placing fake jail cells at sites across Tbilisi.
Opposition leader Levan Gachechiladze said activists would occupy the cages, which symbolize the country turning into a police state.
Saakashvili's opponents blame him for dragging the country into a war with Russia over South Ossetia, resulting in the recognition by Moscow of the province, along with another breakaway republic, Abkhazia. He also faces criticism for failure to carry out democratic reforms promised after the 2003 "Rose Revolution" that brought him to power.
Saakashvili has remained defiant of the resignation demands, saying he would remain in office until his presidential term expired in 2013. The U.S.-educated president cited opinion surveys late last week, saying people wanted dialogue between authorities and opposition, and stability.