President Barack Obama continued his push for a grand bargain, an immigration overhaul and the rest of his agenda in a meeting with House Republicans on Wednesday, but GOP leaders immediately shut the door on any compromises that would include new taxes.

Obama appealed to Republicans to come to the table on a bipartisan budget package — saying he’s willing to back $2 in spending cuts for every $1 in tax revenue — and he argued that reaching agreement on immigration would benefit them politically.

But he also told them balancing the budget over the next 10 years is not on his priority list.

Republicans thanked the president for the rare sojourn to the Capitol and said they hoped that this would start a new dialogue with the president on a range of issues. But many lawmakers said deep suspicions and divisions remain.

GOP leaders immediately held a news conference where they praised the president’s visiting but reiterated their opposition to his tax demands.

“If the president wants to let our unwillingness to raise taxes get in the way, then we are not gonna be able to set differences aside and focus on what we agree on,” House Majority Leader Eric Cantor of Virginia said. Cantor suggested that there are “low-hanging fruit” among spending cuts that both sides support.

And Speaker John A. Boehner of Ohio said he’s heard the president before but that it was good for his members to hear it.

“Republicans want to balance the budget, the president doesn’t. Republicans want to solve our long-term debt problem. The president doesn’t. We want to unlock our energy resources to put more Americans back to work. The president doesn’t. But, having said that, today was a good start, and I hope that these kind of discussions can continue,” he said.

Boehner: Cuts Held ‘Hostage’

Obama, members said, talked about areas where he was willing to trim entitlements, including adopting a stingier measure of inflation and means-testing Medicare — but he explicitly tied those proposals to new revenue. Or, in the words of Boehner, he threatened to hold cuts “hostage.”

“He does not want balance the budget in 10 years and he wants tax increases and he wants new spending, but other than that we are close,” quipped Rep. Darrell Issa, R-Calif.

Obama also faced tough questioning on the sequester-related closing of White House public tours — the president pointed the finger at the Secret Service, causing an audible gasp in the audience.

When the president announced in the middle of the meeting that white smoke had been seen at the Vatican — signifying a new pope — Rep. Billy Long of Missouri seized the occasion. “Billy just blurted out, as Billy does, ‘Does that mean we’ll have tours at the White House?’ and it broke the place up and even the president laughed at that,” Nebraska’s Lee Terry said. Obama reportedly replied that tours of the Vatican were still available.

Other lawmakers said they were suspicious of Obama’s having their best interests at heart on immigration changes, noting that he has a goal of a Democratic takeover of the House.

“I think it would be an extremely foolish thing for Republicans to put 8 million new Democrats on a path to voting,” Steve King of Iowa said.

Other members said that the president — and his staff — have had so little contact with the rank and file that it will take more than one appearance at the Capitol to overcome four years of distrust. Terry said he’s still never spoken to the president, couldn’t name anyone on the White House’s legislative staff and sometimes doesn’t get his letters answered.

Joe L. Barton of Texas said that the president was long on grand vision and often falls short in sending specific legislative proposals to Congress.

“He is not into that,” Barton said.

Mike D. Rogers of Alabama reported that the president diplomatically said “no” to a balanced budget, explaining that eliminating the deficit is not his priority. Instead, the president said he is worried that the deep spending cuts required to balance the budget would slow the nation’s economic recovery.

The final questioner, Georgia’s Tom Price, engaged in an exchange with Obama about whether Republicans can trust Obama, alluding to a recent National Journal story that quoted an anonymous White House official calling Obama’s outreach to Republicans a “joke.”

Price also pressed Obama on when he would be issuing his budget, which is expected two months late around April 8. Price said he thought Obama was holding his budget up to launch political attacks on Congress’ plans. And he noted that the law required the president to go first.

Obama will be back on the Hill on Thursday meeting with Senate Republicans and House Democrats. He met with Senate Democrats on Tuesday.

Humberto Sanchez, Jonathan Strong and David M. Drucker contributed to this report.