The candidacy of al-Barghuthi, who supports armed resistance but says he wants peace with Israel, could shake up the calcified world of Palestinians politics.
By law, elections are to be held by 9 January or within 60 days of Arafat's death on Thursday.
Many believe the popular al-Barghuthi is the only leader capable of unifying squabbling Palestinian factions, reining in resistance fighters and possibly restarting peace efforts with Israel.
However, Israel is determined not to free al-Barghuthi, who is serving multiple life terms for a role in the killings of four Israelis and a Greek monk.
Al-Barghuthi also could represent the best hope for Arafat's Fatah movement to beat down a challenge by the increasingly popular resistance group Hamas, which is considering nominating a candidate.
"When he takes that decision [to run], we will be near him and we will support him," Ahmad Ghunaim, a senior Fatah leader and another member of the young guard said.
"I think he has the best chance in the movement to win the elections."
Al-Barghuthi's wife, Fadwa, said she was unaware of her husband's plans. But his brother Hisham said: "His people around him from the Fatah and Tanzim (Fatah rank and file), want him and if they want him, he is looking to be president."
After Arafat's death on Thursday, Parliament Speaker Rawhi Fattuh was sworn in as the caretaker leader of the Palestinian Authority.
The elections should be held
within 60 days of Arafat's death
Though some officials have talked of amending the law to allow parliament to choose the new leader, Prime Minister Ahmad Quraya said on Saturday elections would be held by 9 January.
Fattuh was to meet on Sunday election officials to decide whether to hold the poll on 7 January or 9 January, Palestinian cabinet minister Saib Uraiqat said.
Ready for elections
Ammar Dwaik, deputy chairman of the Palestinian election commission, said Palestinians were ready to hold their first presidential election since 1996, noting that a voter registration drive was recently completed.
"There is a political will to conduct the elections," he said. "We are ready technically. All we need is a decree to set a date."
Rami Hamd Allah, chairman of the Palestinian election committee, called on the international community to pressure Israel to allow the elections to go smoothly, especially in disputed east Jerusalem.
Israeli police raided registration stations several months ago, preventing most East Jerusalem residents from registering.
A senior Israeli official said Israel had not been formally contacted about the matter.
Release ruled out
Arafat's death raised speculation Israel might release al-Barghuthi in a goodwill gesture, but Israeli Foreign Minister Silvan Shalom ruled that out.
"He will remain in prison for the rest of his life, because he's a murderer, because he's responsible for the killing of so many," Shalom said on Thursday.
Al-Barghuthi intends to run regardless, and will only bow out of the race if Fatah holds primaries and he loses, the person close to al-Barghuthi said on condition of anonymity.
Al-Barghuthi: Attacking Israelis
is justified to end the occupation
Al-Barghuthi, a former West Bank leader of Fatah, has firm street credentials. He spent six years in Israeli jails before being deported in 1987, and was one of the first exiles to return seven years later after interim peace deals with Israel were signed.
He once had close ties with Israeli peace activists, and speaks fluent Hebrew that he learned in prison. But after the outbreak of Israeli-Palestinian fighting four years ago, he said force - including shooting attacks on Israelis - was justified to end Israel's occupation of the West Bank and Gaza Strip.
Pressure on Israel
None the less, al-Barghuthi still says he supports peace. Al-Barghuthi was considering running along with a vice-president - a position that currently does not exist - who would handle the daily administration of the Palestinian Authority while he remains in prison, the person close to him said.
Pressure on Israel to release al-Barghuthi would become intense if he wins.
"If the Palestinians are going to make this work against the really hardline elements, the Islamists and some of the people of Hamas, they're going to have to have a coalition of the young guard and the old guard," former US secretary of state James Baker told CNN on Friday.
"It would be really a very positive step in the right direction if Israel would release Marwan Barghuthi so that he could participate in bringing about this transition," he said.
But a Palestinian cabinet minister said on Sunday the leader of the Palestinian Liberation Organisation (PLO) Mahmud Abbas is the firm favourite to be chosen as the dominant Fatah faction's candidate to replace Arafat as head of the Palestinian Authority.
"Until now we have not started talking officially, neither have we lined up names officially, but everything is pointing in the direction of our brother Abu Mazin," telecommunications minister Azzam al-Ahmad said.
Some officials predict that Abbas
will take over the presidency
Arafat's former top adviser, Nabil Abu Rudaina, also predicted that Abbas, who is also widely known as Abu Mazin, would be Fatah's choice.
"There will be a series of internal Fatah meetings to designate a candidate. It's up to Fatah to decide but Abu Mazin is arguably the strongest candidate," Abu Rudaina said.