Lord Robertson (right): "We are in coalition again"
Lord Robertson will discuss a UK proposal to give Moscow a say in some Nato decisions which he said could lead to a "sea-change" in relations with Russia.
The idea put forward at the weekend by UK Prime Minister Tony Blair - and currently being studied by Nato - does not, however, extend to offering Russia membership of the alliance.
It envisages Russia joining what Lord Robertson described as a special council - one that he said would mark "a huge change, a sea change, in the way we do business".
Lord Robertson's two-day visit to Russia comes amid calls for closer co-operation on both sides.
On Thursday Mr Putin made clear Russia was not asking for membership.
But he said: "We are ready to bring our positions closer to Nato's in many directions, to the extent that the alliance itself is ready for that."
During talks with Russian Defence Minister Sergei Ivanov, Lord Robertson said the West and the Soviet Union had missed an opportunity to work together after World War II and needed now to co-operate to build security for the future.
Taleban threat
He praised Moscow's support for the US-led campaign against terror, describing it as politically crucial.
"Once again we are in a coalition against a common enemy, the common enemy of global terrorism," he said.
But he warned that the threat posed by the Taleban was still real.
"The Taleban is in retreat but is not yet defeated. But they will be defeated and al-Qaeda and Bin Laden will face justice for what they did in New York and elsewhere," he said.
Ahead of the talks, Mr Ivanov said that Moscow was seeking equal status and the "right to a voice" within the Atlantic alliance.
However he warned that on defence issues major differences between Washington and Moscow still remained.
He stressed that the dispute over the proposed US national missile defence system is still far from resolved.
"We are waiting for concrete proposals from the US side," Mr Ivanov said.
The BBC's Steve Rosenberg in Moscow says that in the past all talk of Nato expansion eastwards has been met with fierce words and veiled threats, but that the rhetoric has rapidly changed in recent weeks.
Earlier Lord Robertson visited the southern city of Volgograd, formerly Stalingrad, to lay a wreath at a World War II memorial.
Some one million Soviet soldiers and civilians lost their lives in a gruelling, 200-day battle against Nazi forces at Stalingrad in 1942-1943 that marked a turning point in the war.