David Cameron will tell his Greek counterpart Alexis Tsipras he needs to end the stand-off with the eurozone amid fears the uncertainty could damage the British economy.
The Prime Minister will hold his first face-to-face meeting with Mr Tsipras at a gathering of European leaders in Brussels which is set to be dominated by economic concerns following the victory of the Greek radical anti-austerity party in elections last month.
The European Council summit will also discuss the latest developments in the Ukraine crisis following a diplomatic initiative led by French president Francois Hollande and German chancellor Angela Merkel.
The Ukrainian president Petro Poroshenko has been invited to brief the summit and Mr Cameron is expected to press for the European Union to maintain pressure on Russia's Vladimir Putin to demonstrate that he has changed course.
British officials said Mr Cameron would be in "listening mode" in his meeting with Mr Tsipras, but was keen to hear the details of how he intended to get his country's economy growing and how he would tackle the crucial issue of Greece's debts.
"I expect the Prime Minister to make clear that we expect to see a swift resolution to the current stand-off between Greece and the rest of the eurozone," an official said.
"The Prime Minister's point will be this is not about us taking sides, it's down to Greece and eurozone partners to agree on the best way to move forward to get the Greek economy growing, to bring their debt down and get it on a sustainable footing.
"They need to work on this together but they need to be acting responsibly and coming up with a plan for growth.
"The point from the UK perspective is that the uncertainty around Greece is clearly a risk to our economy. We want Greece and eurozone partners to get on and sort this out."
The talks between Mr Tsipras and his European counterparts will be a chance for the Greek leader to set out how he intends to implement the anti-austerity policies his Syriza party was swept to power on.
The discussions will focus on the detail of the reforms necessary in Greece and the timetable for implementing the measures.
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