The president also suggested he would not be willing to sign other types of tax increases that Democrats have attached to major legislation, including an energy bill, according to numerous officials who attended a closed-door meeting at the White House.
Bush's remarks represented a hardening of the administration's public position in a running veto showdown over Democratic-led attempts to enact legislation that provides coverage for 6 million children who now lack it. The officials who disclosed his comments did so on condition of anonymity, saying they were made in a closed-door meeting.
The president vetoed one children's health bill, and Democrats failed to override him in the House.
His threat to veto a replacement measure that cleared the House last week has led to a hurried round of negotiations among lawmakers in both parties and both houses.
Their goal is to reach a compromise that can command enough votes to gain the two-thirds majority needed in both houses to override the president's veto, if necessary.
The negotiations were private, but in an ominous sign for the White House, Republican leaders said during the day they might defy a White House veto.
House Republican Leader John Boehner of Ohio, asked if he might support a bill that the president would not sign, he replied: "That's always a possibility."
In a similar vein, House Minority Whip Roy Blunt, R-Mo., said he would "have to see the bill" before deciding.
Their comments were the clearest sign yet that even Bush's most loyal House allies are eager for an end to the impasse, which many Republicans see as politically damaging to the GOP.
The White House has said previously it opposes tobacco tax increases that Democrats included in the health care legislation, but only after first detailing numerous other objections. Additionally, the president's press aides have declined repeatedly to say whether he would sign a bill that raised taxes.
Bush supplied somewhat more emphasis in public comments Tuesday.
"You know, they proposed tax increases in the farm bill, the energy bill, the small business bill and of course," the children's health bill, he said of Democrats. "They haven't seen a bill they could not solve without shoving a tax hike into it. In other words, they believe in raising taxes, and we don't.
The vetoed bill would have brought the number of insured children to 10 million. The bill covered kids from families who make too much money to qualify for Medicaid but not enough to afford private health insurance.
The estimated $35 million cost of the measure would be covered by higher taxes on tobacco products, including 61 cents per pack of cigarettes.
Several officials said that in the meeting with Bush, Rep. Joe Barton of Texas asked a question about the president's intentions with the health insurance measure. They said the president responded that he wants his budget director, Jim Nussle, to identify spending cuts to offset the cost of any measure.
They also said Bush appeared to extend his no-tax-increase pledge to other measures. They quoted him as saying that if he signed the tobacco tax increase, it would be difficult to draw the line later on other bills.
Barton could not be reached for comment.
The health insurance bill has emerged as a key flash point between Bush and the Democrats in Congress.
The bill's supporters need to add only about a dozen House Republicans to the 44 who voted Oct. 18 to override Bush's veto. If ongoing negotiations can gain that number, and ideally a lot more, then GOP leaders could embrace the deal regardless of the president's stance, Boehner's and Blunt's comments indicated.
Taxes aside, other sticking points have revolved around Republican demands that poor children gain coverage before others are insured and that strict provisions are included to prevent benefits going to illegal immigrants.
Bush, and most House Republicans, also want to eliminate or reduce participation by adults and families earning more than $62,000 or so.
Boehner said of the disagreement with Bush over how to pay for the program's expansion: "He has his position. The House Republicans have their position."
House Democratic leaders last week said they had addressed many criticisms of the bill and put the revised version to a vote. Most Republicans, encouraged by Boehner and Blunt, rejected the changes, which some called politically motivated.
The bill passed, 265-142. Opposition from 141 Republicans kept it from reaching a two-thirds majority.
The Senate gave the legislation a veto-proof majority from the start, with the enthusiastic backing of senior Republicans such as Sens. Charles Grassley of Iowa and Orrin Hatch of Utah.