"Ukraine has made a bold decision and the United States strongly supports your request."
Vladimir Putin, the Russian president, will also attend the Nato summit in Bucharest set to run until Friday when he will meet with Nato leaders.Nato 'priorities'
Bush will assert his aim to "finish the fight" against Taliban fighters in Afghanistan.
"If we were to let up the pressure, the [fighters] would re-establish safe havens across the country, and use them to terrorise the Afghan people and threaten our own," he said in remarks prepared for delivery at the summit on Wednesday.
In excerpts made public by the White House, Bush said that Nato had outlived the so-called "Soviet threat" it was created to destroy and must now act as "an expeditionary alliance" around the world.
He said: "Our alliance must maintain its resolve and finish the fight in Afghanistan."
The French government appears to have heeded the call.
Francois Fillon, the prime minister, said that the country may contribute "several hundred" more troops to Afghanistan.
Bush also urged Russia to join a planned US missile shield he said was "urgently needed" to thwart a possible threat from Iran.
The president said that a US plan to set up bases in Europe for an intercontinental missile defence shield was not an "anti-Russian" move.
"We're dealing with a lot of history and a lot of suspicion ... I'm hopeful we'll have some breakthroughs, we'll see," Bush said at a news conference in Kiev.
"Obviously we've got work to do to persuade the president and the people around him that the missile defence system is not aimed at Russia."
The US has said it plans to station missiles in Poland and a radar system in the Czech Republic to counter potential threats from "rogue" states such as Iran or North Korea.
Russia says there is no immediate threat to the US mainland from these states and says the US shield is aimed at containing Russia.