Officials say Russia is not being offered Nato membership
He has written to other Nato member states and Russian President Vladimir Putin with new proposals regarding relations between the alliance and Russia.
British officials say Mr Blair's ideas reflect the changing political atmosphere since the 11 September terror attacks in America.
BBC defence correspondent Jonathan Marcus says British officials insist Russia is not being offered Nato membership, nor would it take part in Nato's integrated military command.
Among the proposals is the creation of a new Russia-North Atlantic Council, in which 20 governments would discuss a variety of security issues as equals.
The council would replace the Permanent Joint Council set up in 1997 as a consultative forum between the alliance and Moscow.
Some issues would still be debated within Nato itself.
Security matters
But our correspondent says the new body could deal with security matters that include Russia, such as questions of arms proliferation, peace support operations and so on.
He says the Nato-Russia Permanent Joint Council is seen by both sides as being a product of its times; the immediate aftermath of the Cold War.
The attacks of 11 September have ushered in a whole new relationship and Mr Blair wants to seize the moment and to enshrine this new mood in new institutions, he added.
There have been at least seven discussions between Mr Blair and Mr Putin since the attacks on New York and Washington.
Jonathan Marcus said Mr Blair was hoping for the new Nato-Russia institutions to be established before the alliance's summit in Prague next year, when members may have to tackle the question of the Baltic States' membership of Nato.