Gibbs told reporters the visits were expected in the coming weeks.
After his inauguration on January 20, Obama and his secretary of state Hillary Clinton named George Mitchell as the special envoy for Arab-Israeli peace, a move analysts said signals constant and focused high-level involvement.
Obama's predecessor George W. Bush largely left the peace process to his secretary of state, Condoleezza Rice, after they convened a conference in Annapolis, Maryland in November 2007 to re-launch the negotiations after a seven-year hiatus.
After meeting the Jordanian king, Obama said: "My hope would be that over the next several months ... you start seeing gestures of good faith on all sides.
"I don't want to get into the details of what those gestures might be, but I think that the parties in the region probably have a pretty good recognition of what intermediate steps could be taken as confidence-building measures."
The king was paying his first visit to the White House since Obama became president in January vowing to work for Middle East peace.
"We've had some very fruitful discussions this morning with President Obama," Abdullah told reporters as he began talks with Clinton.
"We're at the State Department now to go over the priorities that Jordan and Arab countries will put in front of themselves of how to bring Israelis and Palestinians to the negotiating table and hopefully open a new chapter of peace and stability in the Middle East and move the peace process forward," he added.
Clinton repeated the Obama administration's backing for a "search for peace that would result from a two-state solution in the Middle East."
She said she and the Jordanian king were in "total agreement" for a two-state solution.
Gibbs, the Obama spokesman, said it was likely the separate visits by Arab and Israeli leaders would take place before Obama is scheduled to visit Normandy for the annual D-Day commemoration in June.
"With each of them the president will discuss ways the United States can strengthen and deepen our partnerships with them, as well as the steps all parties must take to help achieve peace between Israelis and Palestinians and between Israel and the Arab States," Gibbs added.
Obama met both Netanyahu, who was then in opposition, and Abdullah during a visit to their two countries last year after securing the Democratic presidential nomination.
In recent weeks, Obama has made clear to Israel he believes that the path to peace lies in already agreed frameworks ma]codede in the stalled roadmap plan for Israeli-Palestinian peace and the Annapolis agreement.
Two weeks ago, in an address to Turkey's parliament, Obama said that the United States "strongly supports the goal of two states, Israel and Palestine, living side by side in peace and security."
His remarks came after Avigdor Lieberman, the foreign minister of Israel's new right-leaning government, said that the 2007 Annapolis document did not bind Israel though he did accept the roadmap as the basis for progress.