Olmert is "very worried" that both Wednesday's attack and the shooting of seminary students earlier in March this year had been perpetrated by Palestinians from Arab East Jerusalem, the official said.
Police have said they believe that Hussam Dwayat, the 30-year-old driver who was shot after the attack, acted alone and was not connected to any armed Palestinian group.
Defence and legal officials met on Thursday to discuss the demolition issue.
In 2005, Israel pledged to the supreme court that it will not demolish the homes of Palestinians involved in attacks on Israelis.
The pledge came after an Israeli military committee decided that the the demolishing of homes did not act as a detterent against Palestinian fighters.
Legally, however, no formal court order was issued and the committee had itself declared that the practice would be halted.
Several human rights groups had petitioned against the procedure and any renewal of the policy is likely to draw legal challenges.
B'Tselem, an Israeli human rights organisation, on Thursday issued a letter to the Israeli government asking it not to renew the policy of demolishing the homes of those involved in such attacks.
Avi Berg, a spokesperson for the group, told Al Jazeera: "The letter was issued in accordance with what we have heard in the media about [Israeli] intention. We stated again that this practice is illegal in accordance with international law.
"We demanded them not to renew this policy", Berg said.
About 20 people live in the bulldozer driver's family home, relatives say.
'Possibility of demolition'
David Chater, Al Jazeera's correspondent in Jerusalem, said: "The papers are headlining the fact that both the prime minister and the defence minister want to look at the possibility of demolishing the home of the attacker because they say he is involved in a terrorist incident."
Chater said, however, that the Israeli intelligence service has not found any affiliations between the man and [any] Palestinian resistance factions.
"But he did have a criminal record, he spent time in an Israeli prison for rape ... he's also been committed for theft offences.
"Even his father is quoted as saying that his son had a drug problem. We also know that he had a very short temper."
Meanwhile, Israel's vice prime minister has said that he wants Palestinian neighbourhoods excluded from Jerusalem following Wednesday's attack.
Haim Ramon told Army Radio that the Arab neighbourhoods were artificially incorporated into Jerusalem and should be ceded to the West Bank.
"They were annexed in 1967 and we call them Jerusalem, even though there is not one Jerusalemite there," he said.