The process began Nov. 13, and Abbas legal aide Hassan al-Arouri told the Palestinian Ma'an News Agency testing on the body will begin Monday.
The testing will be overseen by French and Palestinian experts, but it will be closed to media and family members.
French authorities initiated an inquiry into the PLO leader's death this year after high levels of radioactive material were found on some of his personal belongings.
Arafat's widow, Suha Arafat, lodged a formal legal complaint for murder. She told CNN she wanted her husband's body exhumed so investigators could be "100 percent sure" the radioactive substance was present.
Other family members have not been as supportive of the exhumation.
"We say openly that our leader, our founder was assassinated by Israel with poison. The overwhelming majority of the Palestinian people is convinced of this," Nasser al-Kidwa, Arafat's nephew and a senior official in Abbas's Fatah group, said Saturday.
Arafat was 75 when he died in a Paris military hospital after suffering a brain hemorrhage and slipping into a coma.
Just before his death, Palestinian officials said he had a blood disorder and digestive problems. Suggestions he had been poisoned had circulated at the time but then-Palestinian Foreign Minister Nabil Sha'ath said he "totally" ruled them out.
A Swiss doctor later said investigators had found high levels of toxic polonium-210 on some of Arafat's belongings, CNN said.
A body fluid stain indicated 180 mega-becquerels per liter of polonium-210; a typical sample would contain 5 mega-becquerels per liter, the doctor said. The fabric of Arafat's clothing, without body fluid, contained less than 10 mega-becquerels per liter. (A becquerel is a measurement of radioactivity.)