Bush's two-day meeting with Merkel, which began Friday at his Texas ranch, came just a few days after Bush met with French President Nicolas Sarkozy.
The meetings marked new chapters in U.S. relations with both nations, which were bruised by sparring that Merkel and Sarkozy's predecessors - Gerhard Schroeder and Jacques Chirac - had with Bush over the U.S.-led invasion of Iraq.
Bush donned a tuxedo for the elegant White House dinner he hosted for Sarkozy, known in France as "Sarko the American." He wore blue jeans to welcome Merkel to his dusty ranch.
"In Texas, when you invite somebody to your home it's an expression of warmth and respect, and that's how I feel about Chancellor Merkel," Bush said, standing at a helipad on the ranch where Merkel landed.
Looking out at the grassy prairie, Merkel, through a translator, replied: "Already a first glance of the area shows us that this is, indeed, a wonderful place to be and a wonderful atmosphere. We have a number of issues that I think we will have now time to discuss."
Their weekend discussions will be dominated by Tehran's decision to defy international demands to halt its uranium enrichment program.
U.S. officials claim Russia and China - two of the five permanent members of the U.N. Security Council - are blocking the U.N. from moving quickly toward a third set of sanctions against Tehran. Merkel will urge Bush to be patient and let the current diplomatic efforts play out.
The view from Berlin is that the U.S. and Europe cannot afford to split with Russia because Moscow is needed to help solve other international problems as well, especially with Russia's parliamentary elections approaching in December, Russian President Vladimir Putin's stepping down as president in May, the U.S.-Russia dispute over missile defense unresolved and talk of enlarging NATO. Russia has expressed concern over NATO's possible farther eastward expansion to include some ex-Soviet states.
Being aligned with Bush carries some political risk for both Sarkozy and Merkel, given the president's unpopularity in Europe. In the past, the German leader has mitigated the effects of Bush's unpopularity by publicly airing some disagreements, such as her belief that the U.S. detention facility at Guantanamo Bay, Cuba, for terrorist suspects be shuttered.
On Afghanistan, Merkel wants to stress to Bush and the American people that Germany will honor its commitments on international crises, and is willing to shoulder even greater responsibilities. Bush, meanwhile, has backed off his criticism of allies in Germany and elsewhere that are not sending their troops to fight in the country's more dangerous south.