Ismail Haniya was named the new chief of Hamas Saturday, a spokesman for the Palestinian Islamist movement in Gaza said.
Barghouti is a popular figure in Palestinian society and is seen as a possible successor to Abbas.
Hamas remains deeply divided from Abbas's more moderate Fatah, which is based in the occupied West Bank.
The PLO is considered the worldwide representative of Palestinians both in the Palestinian territories and the diaspora.
In the decade since, Hamas has fired thousands of rockets at civilian centers in Israel, survived three wars against the Israeli army, and, Israeli officials say, continuously built up its military force.
Hamas is classified as a terrorist organization by the U.S., European Union and Israel.
Haniyeh replaces Khaled Meshaal, who served as Hamas leader since 1996.
The top contenders include former prime minister Ismail Haniyeh and senior Hamas leaders Moussa Abu Marzouk and Mohammed Nazzal.
This week, Hamas published its first new policy document since its founding charter. During his meeting with Palestinian President Mahmoud Abbas at the White House, Trump had said he is committed to working with Israel and the Palestinians to reach an agreement.
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He said the administration was focused on growth, "and growth in the United States is good for the world economy". The gathering paved the way for a meeting of national leaders in Taormina, Sicily on May 26-27.
The next step for Hamas should be to move in the direction of the two-state solution.
Mainstream media sources, including the New York Times, CNN, Al Jazeera, and The Guardian, all reported that Hamas had now accepted the 1967 lines.
Haniyeh made the assertion in his first press statement since his election as head of Hamas's political bureau, which he delivered while taking part in activities to show solidarity with jailed Palestinian hunger strikers.
Hamas leaders have been able to travel overseas from time to time, but only with Egyptian coordination.
Some diplomats have however said Hamas would need to do much more to convince Western countries to end its isolation, noting that the changes announced so far could be merely window-dressing.
In Gaza, where Haniyeh still resides in his home in a refugee camp, some residents saw his election as a sign that could draw attention to the territory's woes.
In a series of interviews conducted via email with the Beirut-based al-Zaitouna Centre for Studies in 2013, Haniya explained that Hamas makes a distinction between political efforts and compromise; "we are not opposed to any diplomatic or political efforts to restore our rights, but we are against bargaining or exchanging our rights".
Haniya and Yassin escaped death in 2003, in a failed Israeli assassination attempt in the form of air raids on an apartment bloc in downtown Gaza where the two men were meeting.
The hunger strike of close to 900 Palestinian prisoners, who demand better conditions from Israel, reached its 22nd day on Monday.
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