U.S. Defence Secretary Robert Gates, speaking in Denmark, also hoped for progress while acknowledging that "the Russians are probably never going to like missile defence.
"But I think that the assurances that we have provided and the mechanisms that we have proposed give them assurance that it's not aimed at them and my hope is that that will lead to positive outcomes both in Bucharest and in Sochi," he said.
The president spent the afternoon visiting monuments, including 11th century St Sofia cathedral and a memorial to victims of the 1932-33 mass famine engineered by Soviet dictator Josef Stalin. He left for Bucharest late in the afternoon.
In his comments to reporters, Yushchenko said Ukraine had made a clear choice on NATO and he saw "no other way forward.
"You will forgive me, but I would not like to see the key, fundamental principle of the Alliance's activity, open doors, to be replaced by a veto for a country which is not even a member."
Prime Minister Yulia Tymoshenko, who also met Bush, hoped for a "yes" in Bucharest and repeated the longstanding policy of all major political forces that a final decision on membership would be subject to a referendum.
Ukraine's bid to secure a MAP, the first stage in the long process of joining NATO, faces low public support at home.
A few hundred protesters defied a ban and shouted anti-NATO slogans in central Kiev. Some 5,000 had massed on Monday.
For many Ukrainians, joining NATO is not a priority - only 30 percent of respondents in the ex-Soviet state support it.
The Regions Party of Former Prime Minister Viktor Yanukovich, long an opponent of the plan, said approval in Bucharest would "spark protests by millions. Those in Europe and across the ocean must be told - Ukraine is not ready for MAP."
France and Germany have resisted granting a MAP on the grounds that both states have yet to achieve political stability and that the process would unnecessarily antagonise Russia.
French Prime Minister Francois Fillon told French radio Paris was opposed "because we think it is not the right response to the balance of power in Europe and between Europe and Russia, and we want to have a dialogue on this subject with Russia".
