"We're going to have to do more work after Sochi," Perino said.
But she added that Bush and Putin were expected to sign an agreement on the "strategic framework" of US-Russian relations, a document some analysts say is designed to cement a legacy for their successors to follow.
Dmitry Peskov, the Kremlin spokesman, was more circumspect, saying: "Let's wait until tomorrow's negotiations."
The missile shield programme includes plans for a radar system in the Czech Republic that would track the sky for any threats as well as plans to base 10 interceptor rockets in Poland.
Al Jazeera's Jonah Hull, reporting from Sochi, said the talks would be a chance for the two presidents to cement ties before their both men's terms run out this year.
"It's an opportunity for these two men to bid a final farewell as presidents," Hull said.
"The major issue is going to be missile defence. The Americans have been trying for some time to assuage Russian suspicions ... but no one is expecting a real breakthrough."
"Let's be friends"
Both Putin and Bush earlier attended the Nato conference in Bucharest, Romania's capital, where Croatia and Albania were formally invited to join the alliance.
At the conference, Putin reaffirmed his opposition to Nato's ambitions for further eastward expansion on Russia's borders and to the defence shield.
But the Russian president's tone did appear more conciliatory, saying at the end of his speech: "Let's be friends, guys."
On Saturday in Sochi, where Putin has a summer home, Bush met Dmitry Medvedev, Russia's next president.
He will hold talks with Medvedev on Sunday, and possibly have a chance to gauge how much power Putin, who will become prime minister, will continue to hold within the Kremlin.