Relations between the two countries fell to a low in 2006 over the death of Alexander Litvinenko, a former KGB bodyguard, who was poisoned by radioactive polonium-210 in London.
"I believe our British colleagues understand that demands to change our constitution are absolutely unrealistic"
Sergei Lavrov, the Russian foreign minister
Russia has refused to extradite Andrei Lugovoi, the main suspect in the case and now a member of Russia's parliament, saying the constitution forbids the extradition of its citizens.
Lavrov said the Russian position on the case "has not changed", after his meeting with Miliband.
"I believe our British colleagues understand that demands to change our constitution are absolutely unrealistic," he said.
But Miliband gave no indication that London would drop its demands for Lugovoi's extradition, saying: "We continue to seek justice for him [Litvinenko]."
Relations between the two countries have also been strained over Moscow's irritation that London is home to a large concentration of anti-Kremlin exiles.
Although some of them are wanted on criminal charges in Russia, Britain has refused to extradite them.
Agreement on Iran
But despite the differences, the two sides sought to show a united front on the issue of Iran's nuclear programme.
Miliband said both London and Moscow wanted a "prompt response" from Tehran on a UN-backed draft deal on nuclear fuel.
"We don't paper over our differences," Miliband added. "But we don't allow them to block co-operation where possible."
Lavrov said Moscow was counting on Tehran to approve the deal, which was drawn up at a meeting last month in Vienna, Austria, with representatives of Iran, Russia, France and the US.
"This meeting ended with an agreement ... which we are counting on all the participants, without exceptions, to approve, including Iran," he said.
Iran has delayed giving a clear response to the draft agreement brokered by the UN atomic energy watchdog, the International Atomic Energy Agency (IAEA), which is seen as a possible solution to the Iranian nuclear standoff.