The complaint was the latest in a growing rift between the Obama administration and the government of Prime Minister Benjamin Netanyahu over how to move forward to achieve peace in the Middle East. Mr. Obama was in Saudi Arabia on Wednesday and is scheduled to address the Muslim world from Cairo on Thursday.
The Israeli officials said that repeated discussions with Bush officials starting in late 2002 resulted in agreement that housing could be built within the boundaries of certain settlement blocks as long as no new land was expropriated, no special economic incentives were offered to move to settlements and no new settlements were built.
The officials spoke on the condition of anonymity so that they could discuss an issue of such controversy between the two governments.
When Israel signed on to the so-called road map for a two-state solution in 2003, with a provision that says its government "freezes all settlement activity (including natural growth of settlements)," the officials said, it did so after a detailed discussion with Bush administration officials that laid out those explicit exceptions.
"Not everything is written down," one of the officials said.
He and others said that Israel agreed to the road map and to move ahead with the removal of settlements and soldiers from Gaza in 2005 on the understanding that settlement growth could continue.
But a former senior official in the Bush administration disagreed, calling the Israeli characterization "an overstatement."
"There was never an agreement to accept natural growth," the official said Tuesday, speaking on the condition of anonymity because of the delicacy of the matter. "There was an effort to explore what natural growth would mean, but we weren't able to reach agreement on that."
The former official said that Bush administration officials had been working with their Israeli counterparts to clarify several issues, including natural growth, government subsidies to settlers, and the cessation of appropriation of Palestinian land.
The United States and Israel never reached an agreement, though, either public or private, the official said.
A second senior Bush administration official, also speaking anonymously, said Wednesday: "We talked about a settlement freeze with four elements. One was no new settlements, a second was no new confiscation of Palestinian land, one was no new subsidies and finally, no construction outside the settlements."
He described that fourth condition, which applied to natural growth, as similar to taking a string and tying it around a settlement, and prohibiting any construction outside that string.
But, he added, "We had a tentative agreement, but that was contingent on drawing up lines, and this is a process that never got done, therefore the settlement freeze was never formalized and never done."
A third former Bush administration official, Elliott Abrams, who was on the National Security Council staff, wrote an opinion article in The Washington Post in April that seemed to endorse the Israeli argument.
The Israeli officials acknowledged that the new American administration had different ideas about the meaning of the term "settlement freeze." Mr. Obama and Secretary of State Hillary Rodham Clinton have said in the past week that the term means an end to all building, including natural growth.
But the Israeli officials complained that Mr. Obama had not accepted that the previous understandings existed. Instead, they lamented, Israel now stood accused of having cheated and dissembled in its settlement activity whereas, in fact, it had largely lived within the guidelines to which both governments had agreed.
On Monday, Mr. Netanyahu said Israel "cannot freeze life in the settlements," calling the American demand "unreasonable."
Dov Weissglas, who was a senior aide to Prime Minister Ariel Sharon, wrote an opinion article that appeared Tuesday in Yediot Aharonot, a mass-selling newspaper, laying out the agreements that he said had been reached with officials in the Bush administration.
He said that in May 2003 he and Mr. Sharon met with Mr. Abrams and Stephen J. Hadley of the National Security Council and came up with the definition of settlement freeze: "no new communities were to be built; no Palestinian lands were to be appropriated for settlement purposes; building will not take place beyond the existing community outline; and no 'settlement encouraging' budgets were to be allocated."
He said that Condoleezza Rice, the national security adviser at the time, signed off on that definition later that month and that the two governments also agreed to set up a joint committee to define more fully the meaning of "existing community outline" for established settlements.
In April 2004, President Bush presented Mr. Sharon with a letter stating, "In light of new realities on the ground, including already existing major Israeli population centers, it is unrealistic to expect that the outcome of final status negotiations will be a full and complete return to the armistice lines of 1949."
That letter, Mr. Weissglas said, was a result of his earlier negotiations with Bush administration officials acknowledging that certain settlement blocks would remain Israeli and open to continued growth.
The Israeli officials said that no Bush administration official had ever publicly insisted that Israel was obliged to stop all building in the areas it captured in 1967. They said it was important to know that major oral understandings reached between an Israeli prime minister and an American president would not simply be tossed aside when a new administration came into the White House.
Of course, Mr. Netanyahu has yet to endorse the two-state solution or even the road map agreed to by previous Israeli governments, which were not oral commitments, but actual signed and public agreements.
In his opinion article in The Washington Post, Mr. Abrams, the former Bush official who was part of negotiations with Israel, wrote: "For the past five years, Israel's government has largely adhered to guidelines that were discussed with the United States but never formally adopted: that there would be no new settlements, no financial incentives for Israelis to move to settlements and no new construction except in already built-up areas. The clear purpose of the guidelines? To allow for settlement growth in ways that minimized the impact on Palestinians."
Mr. Abrams acknowledged that even within those guidelines, Israel had not fully complied. He wrote: "There has been physical expansion in some places, and the Palestinian Authority is right to object to it. Israeli settlement expansion beyond the security fence, in areas Israel will ultimately evacuate, is a mistake."