Macron and Putin said they would also discuss how to de-escalate tensions over Iran.
In his Facebook post, Macron said "progress on many political and economic issues is evident for we're trying to develop Franco-Russian relations".
The protests in Moscow were triggered by authorities barring opposition candidates from running in an election for the city's legislature in September. During the press conference, Macron called for the respect of free speech and free elections in Russian Federation, where authorities have been cracking down on anti-government protesters in Moscow.
He told the French president that he didn't want to see a yellow vests scenario develop in Russian Federation, saying that France had a "concerning level of violence". "We will do everything to make sure the situation remains within the realms of the law", he said.
"I'm glad that they express themselves freely in elections because it reduces confrontation".
"We need to keep up our pressure, our energy to resolve this problem", he added.
"There is a real opportunity to put an end to the conflict that has been going on for five years", he told reporters, saying there had been "real changes" since Zelenskiy came to office in May.
It sparked a war in eastern Ukraine between government forces and Russian-backed separatists which has so far claimed the lives of more than 13,000 people.
Monday's bilateral meeting in Macron's summer home comes ahead of a meeting of the Group of Seven nations in Biarritz, France at the end of the week. "We don't rule anything out".
Russian President Vladimir Putin said on Monday there was no risk of increased radiation levels after a blast at a military site in northern Russia on August 8 that killed at least five people.
The two leaders also disagreed about the conflict in Syria.
French companies have also been pushing to lift European Union sanctions resulting from Russia's actions in Ukraine.