The White House added that a strong resolution on Syria didn't necessarily need to include threats of force but could be "strong in terms of the obligations and expectations that it would impose on the regime to improve the humanitarian access."
Jordan, a key U.S. ally in the Middle East, is hosting almost 600,000 refugees from Syria, and on Saturday, the President announced new loan guarantees for Jordan totaling $1 billion.
The senior official targeted Russia in his remarks, urging the Russians to play a "more constructive role" in finding a solution.
"As long as they remain wedded to the status quo, this is going to be a difficult problem to resolve," the official said, speaking of Russia's continued support for the Syrian regime.
Russia "can't have it both ways," the official said. "They can't say they're in favor of negotiations in Geneva and a transitional government guided by full executive authority and humanitarian access and have a happy Olympics, and then be part and parcel of supporting this regime as it kills people in the most brutal way."
The Olympics remark referred to the Winter Games now being held in Sochi, Russia.
"I don't think any of us have any expectation that they are going to turn on a dime," the official conceded. He said the administration would not exclude the possibility of reaching "a humanitarian resolution" despite Russia's "sorry record" of vetoing U.N. sanction resolutions.
King Abdullah said one of his main concerns, "is the rise of extremism in Syria, the sectarian violence, and if we don't find a solution, the spillover in the region and the effect that will have."
The two leaders considered ways their countries could support what they described as the "moderate opposition" movement, the senior official said.
Elsewhere in the United States, Sen. John McCain, R-Arizona, said "failure is looming" in the U.N.-mediated peace talks.
McCain also criticized Russia.
"The entire strategy for success at Geneva now relies on Russia putting pressure on the Syrian government to engage in a serious and constructive way," McCain said. "Yet, Russia has recently prevented the passage of a much-needed U.N. resolution on bringing aid to desperate Syrian civilians and continues to provide the means by which President Bashar al-Assad carries out his assault on the Syrian people.
"Such actions indicate that the Russian government is simply not a partner for peace in Syria and cannot be relied on to help secure a successful outcome," McCain said.
In Syria, rebels posted a video on social media purporting to show small group of fighters planting explosives in a tunnel underneath what they claim is the Carlton Hotel in Aleppo. The Syrian Observatory for Human Rights said the hotel serves as an anti-insurgency headquarters for government security forces.
CNN could not independently confirm the video's authenticity.
"We are putting our last touches and doing some measuring so we can make sure that we are under the Carlton Hotel," a rebel says in Arabic. "We are going to add the explosives after we finish everything, and you are going to hear good news, Aleppo people."
The video then cuts to an outdoor scene where billowing smoke rises over buildings, an apparent indication that the hotel was blown up Saturday.
The Syrian government didn't immediately respond to the rebels' claims in the video.