"There is going to be enormous speculation," he said in a joint appearance at his ranch with Denmark's prime minister, Anders Fogh Rasmussen.
Bush strongly suggested Iraq's provincial elections in October may mean that bringing more troops home would have to wait until after the voting.
"I think our generals ought to be concerned about making sure there's enough of a presence so that the provincial elections can be carried out in such a way that democracy advances," he said. "I'll wait and hear what they have to say. But yes, that ought to be a factor in their recommendation to me."
On Friday, a senior administration official told reporters during a briefing at the White House, "I fully expect there to be more reductions this year - and so does the president."
Currently planned troop withdrawals are scheduled to bring the U.S. force presence in Iraq down to 15 brigades by July, for a troop total of about 140,000.
The top U.S. commander in Iraq, Gen. David Petraeus, intends to make a report to Bush in early April. Petraeus is expected to recommend the president wait for about four weeks to six weeks after the end of this round of cuts before making further reductions. That would put off any new decision on lowering troop levels again until August or September at the earliest.
Petraeus considers this course necessary to give a clearer indication of the impact of the current cuts on the insurgency, Iraqi government's readiness and security gains.
A chief topic of the talks between Bush and Fogh Rasmussen was NATO's increasingly tough fight in Afghanistan against militants.
The U.S., so far unsuccessfully, has pressed for some NATO members to send more troops to Afghanistan and drop military restrictions that the U.S. says hampers the effort. Bush and Fogh Rasmussen sought to make the case again in their joint appearance before reporters.
"We expect people to carry a heavy burden," Bush said. "If we're going to fight as an alliance, let's fight as an alliance."
Added the prime minister: "We need more troops in Afghanistan."