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Theresa May said she hopes for a frank and open" relationship with Russia after her first meeting with the country's leader Vladimir Putin.
The Prime Minister made the comments in Hangzhou, eastern China, where global leaders are meeting at the G20 summit.
At an earlier press conference with Barack Obama, the pair discussed the situation in Syria, where Russian forces haven supported Bashar al-Assad.
Mrs May said she would urge Mr Putin to allow humanitarian access to war-torn areas in the country.
"While I recognise there will be some differences between us, there are some complex and serious areas of concern and issues to discuss, I hope we will be able to have a frank and open relationship", she said.
Her comments came shortly after President Obama said he was confident the "special relationship" between the US and UK would continue to grow, despite the latter's decision to leave the EU.
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As Barack Obama addressed the G20 summit in Turkey, he reiterated the United State's solidarity with "our French allies", in the wake of the "heinous attacks" that took place in Paris on Friday.
The president said they were a setback back in the fight against Islamic State, but the coalition led by the US was making progress in bringing down the militant group in Syria and Iraq.
"We have always understood that this would be a long-term campaign. There will be setbacks and there will be successes."
"Even as we grieve with our French friends, however, we can't lose sight that there has been progress being made."
Mr Obama said despite some calls some US ground troops in Syria, it is his view that would be a mistake.
"We can retake territory and, as long as we keep our troops there we can hold it, but that does not solve the underlying problem of eliminating the dynamics that are producing these kinds of violent, extremist groups."
The rise in global acts of terrorism undermines international peace and security, G20 officials have said.
A statement released on behalf of world leaders at the G20 summit also warned that incidents like Friday's attacks in Paris endangered efforts to strengthen the economy around the world.
The G20 group remains committed to cutting off the sources of funding for terrorist groups, the statement added, and said terrorism should not be associated with any religion, nationality or ethnic group.
David Cameron has said the "horrific" attacks in Paris and others "underline the threat we all face, a threat to our values, to our way of life and a threat we must defeat together".
Speaking after meeting with Russia's President Vladimir Putin at the G20 summit in Turkey, he said "a very big gap" remains between the West and Russia over tackling the crisis in Syria, but there are signs of a willingness to compromise on all sides.
We need to find a way to work together to bring this fight to an end and to focus on the aim we all share, destroying the evil death cult which is Isil.
On the issue of Russia's airstrikes over Syria, which the Kremlin has said are targeting Islamic State, Mr Cameron said there are some signs they are focusing more on the group and it is hoped this would continue.
However, he also acknowledged that "a few extra bombs and missiles won't transform the situation".
He also stressed the need for a government with the confidence of all Syria's ethnic and religious communities to bring about peace.
Mr Cameron also announced that he will co-host a donors' conference in London next year to raise "significant new funding" to address the refugee crisis Syria and neighbouring countries.
The G20 summit has agreed "important steps" to cut off terrorist financing and counter extremist ideologies, David Cameron has said.
The Prime Minister was speaking at a press conference after discussing counter-terrorism measures with Russian President Vladimir Putin.
He said he had urged Putin to work with the international community to support the transition in Syria away from President Bashar al-Assad, and had made it clear that he believes bombing moderate opposition groups in the country was a "mistake".
Everyone recognises the need for compromise over Assad's future, he added, and said he feels the "gap" on views about what that should be has now "reduced".
He said there are still "very big disagreements" with Putin over both Syria and Ukraine - but said the two countries should work together where there is agreement.