The presidents of the US and Russia have differed in emphasis in voicing concern about the Mid-East crisis at the G8 summit in St Petersburg.
Bush and Putin say personal ties are warm
George W Bush urged Lebanon's Hezbollah to disarm while Vladimir Putin called for a "balanced" use of force.
Israel has been accused of using disproportionate force to secure the release of troops seized by militants.
Mr Bush arrived in Russia's second city on Friday, and the full G8 summit will officially open on Sunday morning.
Speaking at a news conference after the talks, the two leaders said they had reached agreement on global nuclear energy, and fighting nuclear and other forms of terrorism.
On the violence in the Middle East, Mr Bush made clear that he blamed Hezbollah for the fighting.
G8 SUMMIT: 15-17 JULY
The world's seven richest nations - the US, Japan, Germany, France, Britain, Italy and Canada have met annually since 1975
Russia joined in 1998, turning the G7 into the G8
2006 summit to be held in St Petersburg - the first time Russia has hosted the G8
Energy security, infectious diseases and education are on Russia's agenda
Mid-East crisis, Iran, North Korea, and international terrorism are also likely to be discussed
"In my judgement, the best way to stop the violence is to understand why the violence occurred in the first place," he said.
"And that's because Hezbollah has been launching rocket attacks out of Lebanon into Israel and because Hezbollah captured two Israeli soldiers."
"The best way to stop the violence is for Hezbollah to lay down its arms and to stop attacking."
The US leader also called on Syria to "exert influence" over Hezbollah.
Mr Putin said Israel's concerns were legitimate but "recourse to force must be balanced and it must be stopped as soon as possible".
"Escalating violence will not bring any positive result," he said, adding that the G8 leaders would work to secure peace.