The meeting between the two leaders took place on the sidelines of the G8 summit in Germany.
Washington says it wants to deploy 10 interceptor missiles in Poland, and a radar in the Czech Republic as defence against projectiles launched by what it calls "rogue" states such as Iran.
The project has infuriated Moscow which says it will upset the global strategic balance and could be used to launch attack missiles or to spy on Russia. Washington has denied this.
In his comments to reporters, Bush did not directly mention Putin's radar plan, which a White House aide said was new.
"He made some interesting suggestions," Bush said.
It was the two presidents' first one-on-one meeting since Putin criticised the Bush administration at a conference in February, where he accused Washington of trying to force its will on the world and become its "single master".
Stephen Hadley, White House national security adviser, told reporters Putin's idea of using a Soviet-era radar system in Azerbaijan was "a bold proposal".
"The G8 summit is nothing but media attention and wasted money"
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US officials said they would study the offer and discuss it with the Russians.
A Kremlin spokesman said that Putin's suggestion would remove any need for a US radar in the Czech Republic or anywhere in Eastern Europe.
But it was unclear if Bush would consider the idea of dropping the Czech radar, a plan he had vehemently defended.
A Russian news agency quoted Araz Azimov, the Azeri deputy foreign minister, as saying Azerbaijan was ready for formal talks on the joint use of the Qabala radar.
But Yevgeny Volk, head of the Moscow office of the Heritage Foundation, a US think-tank, told Reuters that Putin's proposal was a hoax designed to stop the US basing elements of its anti-missile defence systems in Eastern Europe.
"It looks like an attempt to divert discussion into a side street and make proposals that will hardly be acceptable to the United States," he said.
The Qabala radar, one of the biggest in the world, has operated in the north of Azerbaijan since 1985. It scans the entire Indian ocean, the Middle East and most of North Africa - and can detect missiles launched in those areas.
It is still manned by the Russian military, who lease it from Azerbaijan.