Kerry was in Jerusalem for talks with Israeli Prime Minister Benjamin Netanyahu, who was accompanied by Defence Minister Moshe Yaalon, media reported.
Kerry had to amend his plans to travel from Jerusalem to Jordan amid fierce snowfall and rain storms, opting instead to head later straight to his next stop, Vietnam.
"I'm delighted to be here. I'm on my way to Asia, heading to Vietnam and the Philippines, which is a long-promised trip," Kerry told reporters as he went into his talks with Netanyahu.
"It's been constructive. It's always complicated. And I look forward to having some good talks this morning after meeting with president Abbas last night. We have a lot to talk about, and we will continue this process."
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After meeting Kerry in the West Bank city of Ramallah, the Palestinian political capital, Abbas rejected US proposals for Israel to keep troops in a future Palestinian state along its strategic border with Jordan.
"President Abbas has rejected the ideas presented by the secretary of state," a Palestinian source said on Friday.
Abbas "rejected the ideas on security because there is not a third party".
This refers to a plan by former US national security adviser James Jones under which a third party would deploy along the Palestinian-Jordanian border.
The Palestinian source said that "all disputed issues must be settled".
Israeli and Arab media reports say the plan envisaged by Washington would see Israel maintain a military presence on the border after a peace agreement with the Palestinians.
Israel has always insisted that a continued military presence on the frontier would be vital for its security for some 10-15 years after Palestinian statehood.
In Tel Aviv daily Yediot Aharonot on Thursday, former Israeli national security adviser Giora Eiland wrote that Israel saw a potential threat as not coming necessarily from the Palestinians, but from "other enemies" in the region.
"Israel insists on having contiguous control along the Jordan River, and it seems like the Americans accept this," he wrote.