(CNN) -- The president-elect's senior adviser said that while Barack Obama is monitoring the situation in Gaza closely, "the fact is that there is only one president at a time."
Barack Obama's approach to the Middle East as president will be the subject of much scrutiny.
"There's only one president who can speak for America at a time. And that president now is George Bush," senior adviser David Axelrod said in an interview that aired Sunday on CBS' "Face the Nation."
Israeli airstrikes continued pounding targets in Hamas-ruled Gaza on Sunday, in what Israel said was a response to escalating rocket attacks against southern Israel.
The fighting ignited eight days after a six-month Egypt-brokered cease-fire between Hamas and Israel expired.
Palestinian security sources said Sunday that at least 277 people had been killed and hundreds wounded.
Obama spoke with Secretary of State Condoleezza Rice on Saturday about the violence in Gaza, two transition aides told CNN.
Asked whether Obama takes on a passive role when talking to Rice, Axelrod said, "I would hardly describe him as passive."
"I think they have a good working relationship. And there's a -- I think the calls are largely in the area of fact-finding for him," he said.
"But I think he wants to get a handle on the situation, so that, when he becomes president on January 20, he has the advantage of all the facts and information leading up to that point."
An aide told CNN that Obama initiated the eight-minute phone call.
"The president-elect appreciated the call and the information from Secretary Rice," one aide said Saturday. "He will continue to closely monitor these and other global events."
Obama has pledged to make Middle East peace a priority from the beginning of his presidency. Arab states are calling for a more even-handed approach than what they consider the pro-Israel tilt of the Bush administration, but Israel is expecting Obama to stay true to the pro-Israel posture he showed during the campaign.
Asked if he anticipates that the Obama administration would be just as supportive of Israel as the Bush administration has been, Axelrod said Obama "recognizes the special relationship between United States and Israel."
Axelrod said Obama will work closely with the Israelis, "but he will do so in a way that will promote the cause of peace, and work closely with the Israelis and the Palestinians on that -- toward that objective."
One analyst of the Israeli-Palestinian conflict cautioned against holding the incoming administration to "dangerously high" expectations.
"I think the tone of American politics will change: You're going to get a serious effort on behalf of the new administration," said Aaron David Miller, a public policy scholar at the Washington-based Woodrow Wilson International Center and a former adviser on Arab-Israeli relations to six secretaries of state.
But, he told CNN, "the fact is that unless the Israelis and Palestinians are prepared -- which they're not right now -- to take the political decisions required to overcome the gaps and to sell an agreement to their respective constituents, there's not much a new president, no matter how bold or charismatic he may be, is going to be able to do about that."
CNN's Ed Henry contributed to this report.