Jones, 73, served in the House of the Illinois General Assembly for 10 years until 1983 before joining the state Senate, where he's been ever since. He has four children with his late wife, Patricia.
Jones reportedly told a radio station that he is interested in Obama's seat.
"Yes, I am interested, but I'm not out there promoting it," he said. "I'm letting my record stand for itself as it relates to education, health care ... issues I've worked on."
Jones, who is black, would fulfill the desire of politicians who want to see an African American replace Obama, the only black member of the Senate.
On Tuesday, U.S. Rep. Bobby Rush, D-Ill, called on Gov. Rod Blagojevich to name a black man or woman to the seat. Rush said it would be a "national disgrace" if Obama's seat were not filled by an African American.
U.S. Rep. Jesse Jackson Jr. of Illinois has been actively campaigning for Blagojevich to appoint him to the vacant seat. He told a local newspaper on Friday that he shares Obama's vision for the country and that if appointed, he would seek re-election in 2010. He cited recent polling that suggests he'd perform well statewide.
"So the bottom line is, I believe I'm the best, ideal and most qualified candidate," Jackson told the Daily Herald's editorial board.
Other potential replacements mentioned are state veterans official Tammy Duckworth and Illinois Treasurer Alexi Giannoulias.
While Blagojevich has given no indication of whom he will appoint to the seat, he has said that he wants to make an announcement before the end of the year.