A spokesman said: "We do not support Abu Mazin [Abbas] for the election and we have decided to vote for Marwan Barghuthi to be our candidate for president."
Abbas, the new Palestinian Liberation Organisation (PLO) leader, is expected to be formally named as the candidate for the 9 January election for the mainstream Fatah party that he co-founded with Arafat, of which the Al-Aqsa Martyrs Brigades is an armed offshoot.
A poll conducted last month said al-Barghuthi was the second most popular choice for Palestinian Authority president trailing Arafat, who died on 11 November in a Paris hospital.
Fadwa al-Barghuthi said her husband - widely regarded as the inspiration behind the second Palestinian intifada - was the strongest candidate for the presidency, but that his bid would depend on Fatah.
On Monday, Palestinian Foreign Minister Nabil Shaath said Fatah had reached a consensus that Abbas should be its candidate in the election to replace Arafat.
In June, al-Barghuthi was sentenced in a controversial verdict by an Israeli court to five life terms in prison. He has denied all charges.
Palestinian Prime Minister Ahmad Quraya has ruled himself out of the Palestinian presidential race.