The veteran leader was to meet several foreign ambassadors after holding nearly five hours of talks with the US envoy to Tbilisi late on Tuesday seeking pledges of support from Georgia's neighbours.
The president has warned his opponents, who claim he rigged parliamentary elections 10 days ago and have demanded he should step down, that they will have to answer for their actions before the law.
Up to 300 opposition supporters have kept up a round-the-clock vigil and are expected to be reinforced by several thousand others on Wednesday.
"Those who have insulted our people will answer before the law. The radical opposition will answer for their actions," Shevardnadze said late on Tuesday.
More Georgians are expected to join
in round-the-clock protests
He did not specify what actions he was referring to, but the message to the protest leaders was clear and was backed up by Georgia's law enforcement agencies, which have transferred units of interior ministry troops to Tbilisi.
The protests were sparked by a 2 November parliamentary election which several opposition parties said was rigged by the government.
They have snowballed into the biggest demonstrations seen in this turbulent former Soviet republic since the 75-year-old Shevardnadze came to power a decade ago.
A regional chieftain from western Georgia, who has emerged a key ally of Shevardnadze, was to meet with Azerbaijan's President Ilham Aliyev in Baku after seeing the Armenian leader Robert Kocharian.
Aslan Abashidze "is looking for support at the request of Shevardnadze from these countries," said Nino Burjanadze, the speaker of the outgoing parliament and one of the protest leaders.
There are very substantial minorities of Armenians and Azeris in Georgia whose support could prove crucial to the president's survival.
The United States late on Tuesday urged Shevardnadze to keep talking to the opposition.
"Those who have insulted our people will answer before the law"
"I passed on the position of my government, which is that negotiations must continue between the Georgian government and the opposition," said US ambassador Richard Miles.
The US provides large sums in aid as well as military assistance to the impoverished country in the strategic Caucasus mountains region.
Shevardnadze had a meeting on Sunday with Georgia's three main opposition leaders, but the talks broke down and the political crisis deepened.
At the weekend, Shevardnadze had appeared helpless in the face of the protests, but had bounced back after marshalling his support at home and receiving an endorsement from Russian President Vladimir Putin.