Britain and Russia are split on whether Syrian president Bashar al Assad should stand down amid the bloodshed in the country.
Foreign Secretary William Hague called for the leader to quit following international outrage at the weekend massacre in Houla which killed at least 100 civilians, including dozens of children.
But Russian foreign minister Sergei Lavrov said regime change in Damascus was "not the most important thing".
Speaking at a joint press conference in Russia, Mr Hague repeated his demand for Assad to hand over power, saying: "We have a long-standing view about this in the UK. We have said all the way back from last August that finding a solution to this involves him standing aside."
But, sitting next to Mr Hague, Mr Lavrov rejected the call, telling reporters: "For us it's not the most important thing who is in power in Syria. For us the main thing is to provide for the end of all violence."
The pair agreed Syria must implement a six-point plan for peace drawn up by former United Nations secretary general Kofi Annan, who has been appointed a peace envoy to Syria.
Russia, which is a long-term ally of Syria with a crucial military base in the country, has sometimes acted as a block to international action against the Assad regime over the past year.
But both Mr Hague and Mr Lavrov said they were in agreement about the importance of the implementation of Mr Annan's plan, which envisages a truce, withdrawal of troops and heavy weapons from cities, deployment of a UN monitoring force, and dialogue between the government and opposition aimed at a Syrian-led political transition.
Mr Hague said world leaders "look to Russia as having a particularly powerful role in being able to exert that additional pressure" on Assad's regime. But Mr Lavrov insisted: "We do exert pressure on the Syrian government daily," and called for the Syrian political process to run its course "without any foreign interference".
Downing Street said PM David Cameron intends to discuss the situation in Syria in phone calls with President Francois Hollande of France and UN Secretary General Ban Ki-moon.