The Obama transition team, which is working to put a cabinet in place before the transfer of power on January 20, has made no comment on the reports.
Obama has already turned to many former members of the administration of Bill Clinton, the former president and Hillary's husband, for help in his transition to power, including Rahm Emanuel, his chief of staff, and John Podesta, the co-chairman of his transition team.
"She probably knows every major foreign leader. There's already a relationship where she can sit and talk directly about the problems that exist either on a bilateral or multilateral basis," Jon Corzine, the Democratic governor of New Jersey, said.
Richardson has an extensive foreign policy resume, serving as ambassador to the United Nations under Bill Clinton and has conducting diplomacy for the US in such hot spots as Sudan and North Korea.
He also served in Clinton's cabinet as energy secretary and angered his former boss when he endorsed Obama after ending his own primary campaign this year.
For 16 months, Clinton and Obama fought a sometimes bitter battle to secure the Democratic nomination for the presidency.
In June, she eventually conceded to Obama and urged her supporters to back him in the race for the White House.
Ahead of the Democratic convention in August there had been speculation that Obama might have tried to reach out to disgruntled Clinton supporters by inviting her to join him as his vice-president.
Obama instead chose Joe Biden, a fellow Democratic senator and a foreign affairs veteran - a policy area where Obama was seen as lacking experience.
The role of US secretary of state is the highest-ranking post in the president's cabinet and is fourth in line to succeed the president.
Democrats have said that Clinton and Richardson are not the only candidates.
John Kerry, the Massachusetts senator who lost the 2004 presidential election to George Bush and Chuck Hagel, the retiring Republican senator of Nebraska who backed Obama over his own party's candidate, were also reportedly being considered.
"You need someone who can be taken seriously internationally, who has that kind of experience and policy ability, but also has the gravitas to make people sit down at the table," Liz Chadderdon, a Democratic strategist, said.
Clinton's husband, former president Bill Clinton, appointed Madeleine Albright as the first woman to become US secretary of state in 1996.