European diplomats have called the meeting at a lavish estate on the outskirts of Mr Putin's hometown of St Petersburg to get reacquainted with the Russian leader since his return to power.
But the crisis in Syria may overshadow talks on trade and other issues at the twice-yearly summit.
Moscow has blunted Western efforts to condemn Syrian President Bashar al Assad and push him from power.
Both Russia and Europe still have hope in UN peace envoy Kofi Annan's UN-backed peace plan to end 15 months of bloodshed that Western nations blame on Mr al Assad.
But EU nations want Russia to press the Syrian leader to withdraw weaponry and halt attacks as demanded by the plan, and want him to step aside to make way for a political transition.
"We need to make sure that Russia is using fully its leverage in convincing the regime to implement (the plan)," an EU official said, declining to be named.
"The Russian side has certainly not been very helpful in finding solutions in terms of a political way out."
Russia says it is not protecting Mr al Assad but that the Syrian leader's exit cannot be a pre-condition for political dialogue.
EU foreign policy chief Catherine Ashton, who met Russian Foreign Minister Sergei Lavrov on Sunday, said in a statement: "Russia's role is crucial for the success of Annan's plan."
She added that the EU wanted to "work closely with Russia to find a way to end the violence and support" the plan.
The statement said she spoke to Mr Annan by phone on Sunday and that they agreed the crisis had reached a "critical point".
At least 39 people were killed across Syria on Sunday, the Syrian Observatory said.
Fighting was reported in Damascus province, the northwestern province of Idlib and near the northern city of Aleppo, it said