"We face common challenges. I look forward to working with you all to meet those challenges," Mr. Bush, flanked by Cabinet officials, said at the White House.
Mr. Bush said that the tax changes in his budget proposal — including the elimination of taxation on corporate dividends and making permanent the end of the so-called marriage penalty — would encourage growth. He said his budget plan includes $400 billion in grants to states, a 9 percent increase.
The governors are in Washington for the National Governors Association's winter meeting.
Mr. Bush won loud applause when he called for the states to embrace his program — known as No Child Left Behind — to improve education. Some governors have complained that the program demanded increased spending but the federal government hadn't provided any additional funds.
"I want to work with you on education," the president said. "If you have any questions about No Child Left Behind, I'll be glad to answer your questions."
The day before, at Sunday's formal dinner at the White House, Mr. Bush recalled the advice his counterparts gave him when he was governor of Texas. He listed the responsibilities he and the governors shared — homeland security, job creation, affordable health care. "We'll meet them together," he said.
CBS News Correspondent Mark Knoller reports Mr. Bush also noted that he feels a special kinship for the governors and said he's grown especially close to the governor of Florida. "The man is like a brother to me," he said.
States are in their third year of shrinking revenues and budget cuts, with some $30 billion of additional cuts overall needed in the next few months. Next year looks even worse, with governors predicting an $82 billion gap between spending demands and their weak tax revenue.
Plans for the meeting with Mr. Bush caused partisan friction among the governors, as Democrats complained the administration was setting limits on the usually free exchange of ideas. They said the White House planned to allow just two questions that had to be approved before the gathering.
White House officials promised a wide-ranging exchange, and as Monday's meeting began Mr. Bush repeatedly said he wanted to work with the governors. As the meeting ended, he asked "Questions?" and then stopped as reporters were still filing out of the room before the meeting began. "Not yet, get the press out."
Before the meeting began, Republican Dirk Kempthorne of Idaho, vice chairman of the National Governors Association, said he was confident the meeting would be open and constructive. He said he was asked to give the administration his question beforehand "so no one's caught flatfooted."
Kempthorne said the constraints were more about the demands on Mr. Bush's time than anything else, and that he expected the limits would be put aside. "He's going to want to talk with the governors. That's his style," he said.
Republican Bill Owens of Colorado complained that Democrats were politicizing the annual sit-down. "When Republicans met with President Clinton, we didn't whine over what points of access we had," said Owens, who heads the Republican Governors Association. "We treated the president with respect."
On Sunday, a proposal to ask Washington for "immediate federal fiscal relief" to help states cover their budget shortfalls and spur their economies was set aside on a party-line vote, with Republicans opposed to the request and Democrats in favor.
"You can't just keep printing money," said Republican Jeb Bush of Florida, the president's brother. "That policy position ... was the position of big government."
Over the next two days, governors will consider other proposals on:
Homeland security, where they hope to receive more of the $3.5 billion promised in the current year, as well as an additional infusion of cash in the coming year.
Medicaid, the $240 billion state-federal program that provides health care to the poor. They're struggling to find a balance between seeking flexibility to reshape the program and asking for more money to shore up the state burden. Governors say Medicaid increases are driving much of their budget problems.
Education, where governors are considering a request to ask Congress to increase financial support to state and local education agencies to cover increased costs incurred by Mr. Bush's No Child Left Behind education program.
Later Monday, governors were to hold a detailed discussion on Medicaid with Health and Human Services Secretary Tommy Thompson.
The annual governors meeting traditionally strives to reach a consensus and present a unified front from the states, but this year the financial pressures are adding to the difficulties.
"We need a carefully tailored jobs incentive package now, not later," said Democrat Gray Davis of California, questioning the value of the Bush budget proposal.
Bipartisan agreement, several governors said, would be more difficult to find this time around.
"Usually, there's a couple things we can all get behind," said Republican John Rowland of Connecticut. "I don't think so this time. Not just yet."