Nelson said that Assad had indicated a willingness to cooperate with the US and the Iraqi army to close or control the border between the two countries.
Washington accuses Syria of allowing weapons and fighters to cross the border into Iraq, something Syria denies.
Bush, who has resisted pressure to hold talks with Syria, as well as Iran, as part of a new approach to ending violence in Iraq, issued a statement on Wednesday calling on Syria to "to immediately free all political prisoners".
'Mechanisms for cooperation'
The official Syrian news agency, SANA, said Nelson and Assad discussed the instability in Lebanon, Iraq and the Palestinian territories.
It said: "There was mutual interest to activate dialogue and putting in place mechanisms for cooperation."
The White House said members of Congress should not be going to Syria.
Tony Snow, a White House spokesman, said: "I think it's a real stretch to think that the Syrians don't know where we stand or what we think. We have made it clear and we will continue to make it clear.
"What has not happened is the appropriate response by the Syrians in terms of their adventurism within the region, especially with regard to Lebanon, their continued support and housing of terrorist organisations."
'Policy of isolation'
Another Democratic senator announced that he too was going to Syria for a meeting with Assad next week. Connecticut's Chris Dodd said in a statement that he originally planned to make the trip in April but at that time heeded a senior administration official's warning not to go.
This time, he said he was going despite state department opposition because "it is clear that a policy of isolation has not altered Syria's behaviour and the instability in the region is greater, not less, than it was eight months ago".
Last week, the Iraq Study Group urged the Bush administration to consider direct talks with Syria and Iran.
The White House has so far rejected such contacts.
Nelson said the state department had sought to discourage him from meeting Assad "because their position is that they did not want to have any contact with Assad".
He said: "I replied that in light of the Baker-Hamilton study group report, that it was obvious that we were going to have contacts, and that remember what the goal was... to stabilise Iraq."
Political sources in Damascus said other US senators would visit before the end of the year, including Massachusetts Democrat John Kerry and Pennsylvania Republican Arlen Specter.
Next month, Democrats will take control of the US Congress from Republicans.