Putin has forcefully objected to U.S. plans for deploying an anti-missile system in Europe, and he has dismissed British attempts to extradite a murder suspect from Russia as stupid.
"I've always had good relations with President Putin. We want good relations with Russia, but that can only be done on the basis that there are certain shared principles and shared values," Blair said in the House of Commons shortly before leaving for the Group of Eight summit in Germany.
"And the consequences if there aren't is not that - there is no point in making hollow threats against Russia - the consequence is that people in Europe will want to minimize the business they do if that happens.
"I personally think a close relationship between Europe and Russia is important, but it will only be a sustainable relationship if it is based on those shared values," Blair said.
In an interview broadcast earlier Wednesday, Blair said the missile plan "has always been about the danger of rogue states."
"The truth of the matter is that, for all sorts of reasons, it is not something that is really about Russia at all and yet suddenly it is put up by Russia in this way, in quite a confrontational way," he said in the interview with the British Broadcasting Corp.
"Now I think the sensible thing, and this is what I'll do certainly when I meet President Putin, is just to have a frank conversation about the state of the relationship between not simply Britain but Europe and Russia," said Blair, who will be attending his last G-8 summit.
Blair added: "I don't really think that in the end it will be in the long-term interest of Russia to have a relationship with Europe or with the Western world that is scratchy and difficult."
On Monday, Putin reiterated the Russian position that its constitution forbids extradition, and he criticized British officials of ignoring that. "From whatever side you look at this problem, there's only stupidity," Putin said in an interview
Blair said he hoped to resolve the issue.
"We know what issues the Russians have there but we can't have someone murdered on British soil in that way and nothing happen, so it is a discussion we will have to have," Blair said.
Despite tensions with Russia, Blair said he did not believe a second Cold War was brewing.
"I don't think that the danger is that you get a fresh Cold War, but I think, if Russia wants the right relationship with Europe, with America and so on, then its' got to be on the basis of certain shared values and principles, and that means that there is a certain way that you raise concerns and issues," he said.
Turkey has found itself in a circle of countries subject to US and European sanctions. Are they dangerous for Ankara? What is Turkey going to do in response?