Mr Sharon is reported to have told US officials that Washington should not in future invite Mr Arafat - who was a regular visitor during the administration of Mr Bush's predecessor, Bill Clinton.
A BBC correspondent in Washington said Mr Bush apparently lent a sympathetic ear to the visiting Israeli leader.
Mr Sharon asked for US support in pressing the Palestinian leader Yasser Arafat to end violent protests against Israel's occupation of the West Bank and Gaza Strip.
"The first thing and most important one is to bring security to the citizens of Israel. Once we reach security, we will start with our negotiations to reach a peace agreement," he said.
Mr Sharon is reported to have told US officials that Washington should not in future invite Mr Arafat - who was a regular visitor during the administration of Mr Bush's predecessor, Bill Clinton.
Mr Bush's spokesman would not confirm or deny whether the Palestinian leader would be welcome, but said: "We do have a special relationship with Israel and the president is pleased to meet with the prime minister".
Mr Sharon also explored the possibility of a boost in the billions of dollars of US military aid which Israel receives, as promised by former President Clinton as part of a now-abandoned plan for a peace treaty with the Palestinians.
Cordial reception
BBC Middle East analyst Roger Hardy says Mr Sharon has received a remarkably cordial reception in Washington for a man with such a controversial reputation.
Mr Bush recalled how Mr Sharon had taken him on a tour of Israel before either believed they would win their respective elections and remarked that the Israeli candidate had ran a much better campaign.
Observers agree that the meeting's main purpose was to establish a good rapport between the two men, especially as Mr Bush has indicated an intention to be far less involved in detailed Mid-East peacemaking than Mr Clinton.
More than 400 people, mostly Palestinians, have been killed since September last year when the Palestinian uprising began in protest at Israel's conduct in the problematic Mid-East peace process.
Settlement controversy
Mr Sharon's meetings coincided with an announcement by the Jerusalem municipal authorities that initial approval had been given for nearly 3,000 more homes to be built at a controversial Israeli settlement Har Homa.
A US State Department spokesman refused to say whether Washington viewed the construction - on land occupied since 1967 - as a violation of international law.
"We don't think it contributes to peace and stability," spokesman Richard Boucher said. "That's about as much as I can say at this moment."
Palestinian parliament speaker Ahmed Qorei said the US should put more pressure on the Sharon government to accept UN resolutions as the basis of a peace settlement.
Another politician, Hanan Ashrawi, described Mr Sharon's Washington visit as an "emerging love-fest, as though the United States is suffering collective amnesia about Sharon's previous crimes".
Arabs accuse the new Israeli premier of being either directly or indirectly involved in a string of atrocities committed against civilians during his career in the Israeli army and as defence minister - including the massacre of Palestinians by Israel's Christian allies in two refugee camps in Beirut in 1982.