Official results are expected early on Monday, but a senior LDP official acknowledged that the party was heading for a "historic defeat".
"The predictions by the media were shocking. We had doubts, but now I think they are becoming a reality," Yoshihide Suga, deputy chairman of the LDP's election strategy council, was quoted as saying by Reuters news agency.
Mr Hatoyama, who is almost certain to lead the next government, is the wealthy grandson of the founder of Bridgestone tyres, whose other grandfather was a former LDP prime minister.
He has promised to boost welfare, reform the bureaucracy and seek a more balanced relationship with the United States.
Whatever that means (italics mine)- From the BBC Web Page.