The two leaders in the Black Sea resort of Sochi.
Putin's spokesman Dmitry Peskov says Assad informed Putin of his decision to "send a delegation to the U.N." to discuss reforming the country's constitution.
According to Putin, "much has already been done within the Astana process".More news: Congress' midnight drama at Supreme Court
"Much has also been done within the framework of the Syrian National Dialogue Congress held in Sochi".
"The next task, of course, the economic recovery and humanitarian aid for those people in a hard situation", Mr Putin added. "But with you and other partners and friends, we will continue to make strong progress in the peace process", Mr Assad was quoted as saying in the statement.
"Of course, after this military success additional conditions have been created to restart a full-format political process", he said.
Speaking Thursday at a meeting with the top military brass in the southern Russian city of Sochi, Putin said the Russian air forces would receive 160 new aircraft this year and the army is to get 500 new armored vehicles and artillery systems.More news: Levin breaks down the FBI's secret 'Crossfire Hurricane' 'cabal' against Trump
According to a Kremlin statement, Mr Assad said "stability is improving" in Syria, "opening the door to the political process we started some time ago".
Russian Federation has been a key ally of Assad throughout the seven-year Syrian civil war when many other countries, including the United Kingdom, wanted to remove him from power.
The Astana peace talks were launched on January 23-24, 2017, with the aim of putting an end to violence and improving the humanitarian situation in war-torn Syria.More news: Russia Opens Longest Bridge in Europe