But the British prime minister was more circumspect, telling Aljazeera that the security situation was still difficult and stressing the right of the elected Iraqi government to use maximum force to end what he described as "terrorism".
He added that foreign troops would leave Iraq when asked to do so by an elected Iraqi government.
The White House played down any talk of schedules.
Nevertheless, the rhetoric in Baghdad reinforced suggestions by Iraqi political sources that the new government was stepping up efforts to reach out to moderates among Sunni fighters.
Nuri al-Maliki, the prime minister, keen to show Iraqis the days-old national unity government can deliver independence, said his forces could be running all of Iraq by late next year.
"Our forces are capable of taking over security in all provinces in Iraq within a year and a half," he said in a brief written statement after talks with Anders Fogh Rasmussen, the Danish prime minister, another foreign leader with troops in Iraq.
When Blair was in Baghdad two days ago, al-Maliki said all but the capital itself and al-Anbar could be under Iraqi control by the end of this year. Bush, the US president, and Blair played that down but said the 150,000 foreign troops would go once Iraqi forces were ready.
Des Browne, the British defence secretary, continued that theme on Wednesday, saying a planned handover of responsibility soon for at least one of the four southern provinces controlled by Britain would not mean an immediate reduction in British troops.
"We want Iraq to take full responsibility for its own security ... But we cannot go too fast if we want the progress we are making to be sustainable," he said.
Tareq al-Hashemi, the Iraqi vice-president, said he hoped Bush and Blair would reverse their position that withdrawal was conditional on Iraqi capabilities and not a fixed date.
He said he and Blair had discussed a possible compromise, involving setting a date for the completion of Iraqi forces' training rather than for foreign troops' withdrawal, and said it could draw some fighters to the negotiating table.
"The suggestion made two days ago was to combine the need for a timetable with the condition that Iraqi forces must be rebuilt. There would be a US and British commitment to rebuilding them according to a set timetable," al-Hashemi said.
"Blair promised to study this with Bush in their coming meeting ... But no, there is no withdrawal timetable for now."
White House spokesman Tony Snow ruled out any public change of tack on Wednesday: "I do not believe that you're going to hear the president or the prime minister say we're going to be out in one year, two years, four years ... I don't think you're going to get any specific prediction of troop withdrawals."