The website of Haaretz newspaper said Sharon was paralysed in half of his body.
It quoted sources in Sharon's office as saying they were hoping for a "miracle" to save the prime minister, though doctors said the chances of recovery were "very low".
"A significant stroke means that there has been extensive damage," professor Shlomo Moryussef of the Hadassah hospital in Jerusalem told reporters.
The director of the hospital said Sharon has been taken for surgery after suffering a cerebral haemorrhage.
"The diagnosis is cerebral haemorrhaging. In light of this,
the prime minister has been taken to the surgical ward," Dr
Shlomo Mor-Yosef told reporters.
Shmuel Shapira, a doctor at the hospital, told the private
Channel 2 television network that Sharon had been "taken to an operating room to drain the blood" from his brain.
A hospital spokesman said that the operation could last "two to three hours" and was being conducted by two neurosurgeons.
"The imaging test showed he had a haemorrage inside his brain," he told reporters.
Sharon's chief spokesman Ranaan Gissin told reporters that the prime minister's condition was "going well."
"The operation is going well and his condition is stable,"
Gissin told reporters around an hour and a half after the start of the operation.
"Of course I am concerned but I hope that Sharon will win this fight like so many he has won before," he added.
Yisrael Maimon, the Israeli cabinet Secretary, said Sharon's authorities have been transferred to his vice-premier, Ehud Olmert.
"Given that the prime minister requires treatment, I have spoken to the attorney general (Menachem Mazuz) and deputy Prime Minister Olmert and it has been decided that Mr Olmert will have powers transferred to him."
Channel 2 TV had earlier reported Sharon was suffering from paralysis in his lower body.
Analysts on Israeli TV stations said his life could be in danger. Channel 2 TV said he was taken into the hospital on a stretcher.
Sharon had been scheduled to undergo a heart procedure to repair a small hole on Thursday at the hospital.
He was taken by ambulance, a drive of more than an hour from his ranch in the Negev Desert in Israel's south, instead of by helicopter.
"A significant stroke means that there has been extensive
Professor Shlomo Moryussef of the Hadassah hospital
Sharon was at his ranch when he suddenly "felt ill", a close aide said on condition of anonymity.
On 18 December, Sharon was taken to Hadassah Hospital from his office after suffering a mild stroke. Doctors said he would not suffer long-term effects from the stroke, but they discovered a birth defect in his heart that apparently contributed to the stroke.
Israeli security men stand guard
while Sharon undergoes a surgery
Since the stroke, Sharon has been receiving blood thinners to try to prevent a recurrence of the clotting that caused the stroke.
Sharon, who will be 78 next month, has cut down on his foreign travels in recent years but still maintains a punishing workload.
He has been under huge stress in recent weeks, not least with the prospect of his son Omri facing prison after pleading guilty to charges of providing false testimony and falsifying documents following an investigation into allegations of illegal financing of one of his father's leadership campaigns.
News also emerged on Tuesday night that Israeli police are to
examine computer data which they believe will show Sharon's family received an alleged three million dollar bribe from Austrian financier Martin Schlaff.
Prayers for Sharon
As news of Sharon's stroke spread, world leaders as well as those in Israel wished him speedy recovery.
George Bush, the US president, said he was praying for the prime minister's recovery.
"Laura (Bush) and I share the concerns of the Israeli people about Prime Minister Ariel Sharon's health, and we are praying for his recovery. Prime Minister Sharon is a man of courage and peace. On behalf of all Americans, we send our best wishes and hopes to the prime minister and his family," Bush said.
Political leaders in Israel also united in prayers for Sharon.
Benjamin Netanyahu, former Israeli prime minister and a rival of Sharon, was among the first to offer his wishes for a full recovery.
"Like the entire people of Israel, I also pray for the well being of Prime Minister Ariel Sharon," Netanyahu said in a statement.
His wishes were echoed by Moshe Katsav, the Israeli president, who said that "along with all the Israeli people I pray for the well being of the prime minister and wish him a speedy recovery."