By David Brunnstrom

CAIRO (Reuters) - The United States and Egypt are returning to a "stronger base" in bilateral ties despite tensions and human rights concerns, U.S. Secretary of State John Kerry said on Sunday after talks with his Egyptian counterpart.

"Egypt remains vital ... to engagement and stability in the region as a whole," said Kerry, who held the first bilateral strategic dialogue since 2009.

"There are obviously circumstances where we have found reason to have grave concern and we have expressed it very publicly," he said at a news conference with Foreign Minister Sameh Shukri. "But we have multiple issues that we need to work on simultaneously."

U.S.-Egyptian relations cooled after Islamist president Mohamed Mursi was ousted in 2013 by the military amid mass protests against his rule.

Cairo remains one of Washington's closest allies in the Middle East, an increasingly crucial role in a region beset by turmoil in Syria, Iraq, Yemen and Libya.

Kerry also lobbied for last month's nuclear deal between world powers and Iran, which has been met with scepticism by Washington's Arab allies.

"There can be absolutely no question that the Vienna plan, if implemented, will make Egypt and all the countries of this region safer than they otherwise would be," he said.

Kerry said Iran's role as the "number one state sponsor of terror in the world" made the deal even more important. "If Iran is destabilizing, it is far, far better to have an Iran that doesn’t have a nuclear weapon than one that does," he said.

"NO MAJOR DISAGREEMENTS"

Shukri said Cairo had no major disagreements with Washington, only "differences in points of view over some issues, which is natural".

Washington supports former general Abdel Fattah al-Sisi, who led Mursi's overthrow and was elected to succeed him, but has cautiously criticized Egypt's human rights record.

Following Mursi's ouster, Cairo cracked down on the now outlawed Muslim Brotherhood. Security forces killed hundreds of Brotherhood supporters and jailed thousands more, and later pursued liberal activists.

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Egypt says the Brotherhood is a terrorist group. The Interior Ministry denies allegations of widespread human rights abuses.

Kerry later met Sisi and a senior State Department official said he stressed "the importance of press freedom and the protection of peaceful dissent, stressing that free participation in the political process is essential to help stem the growth of violent extremism".

In one such rights case, a Cairo court on Sunday postponed its verdict in the retrial of Al Jazeera television journalists accused of aiding a terrorist organization, a reference to the Brotherhood.

Shukri said no journalists in Egypt were in jail over their reporting. The Committee to Protect Journalists said a prison census it conducted on June 1 found at least 18 Egyptian journalists were in jail for their reporting.

Cairo and Washington have been exploring ways to expand their security relationship. In late March, U.S. President Barack Obama lifted a hold on supplying arms to Cairo, authorizing deliveries valued at over $1.3 billion.

The United States delivered eight F-16 fighter jets last week to Egypt, which is battling insurgents in the Sinai Peninsula who have killed hundreds of soldiers and police since Mursi's removal. The most active group there is an affiliate of Islamic State.

Also on Sunday, demonstrators in central Cairo burned American and Israeli flags in protest against the death of a Palestinian toddler who died in a suspected Jewish extremist arson attack on his home in the occupied West Bank on Friday.

Kerry will travel on to Qatar to discuss the fight against Islamic State with other Arab leaders and seek to reassure them about the Iran nuclear deal.

(Additional reporting by Omar Fahmy; Writing by Yara Bayoumy and Shadi Bushra; Editing by Tom Heneghan)