The university has a 17-acre site near its Upper West Side Manhattan campus that could house the Obama library complex, the newspaper and Crain's Chicago Business reported.
Columbia -- whose alumni include three U.S. presidents -- has informed Obama's inner circle it's prepared to do what is necessary to win the presidential library after Obama leaves office in January 2017, Crain's said, citing multiple sources, some with close White House connections.
"I think New York is going to push hard," a source told the Sun-Times.
Columbia President Lee Bollinger, a lawyer like Obama and a noted First Amendment scholar, is directly involved in the campaign, Crain's said.
"While it is premature to comment on a library selection process that has yet to be initiated, Columbia looks forward to learning more about the objectives that will inform President Obama's ultimate decision when that information becomes available," Columbia Vice President Robert Hornsby told the Sun-Times.
The White House is expected to announce the formal selection process for the library and museum and the creation of a foundation to house the effort after Obama's Jan. 28 State of the Union speech.
A decision on where the library will be housed is expected after the midterm elections in November, Crain's said.
The Chicago media reported the interest because the University of Chicago was the presumed front-runner for the repository for Obama's presidential papers, records, collections and other historical materials.
The university -- affiliated with 89 Nobel laureates, including 10 current faculty -- has been working with first lady Michelle Obama's former chief of staff, Susan Sher, on its library proposal, the Sun-Times said.
But both newspapers said Columbia's involvement changed the dynamic.
The Sun-Times said Chicago Mayor Rahm Emanuel, Obama's former chief of staff, might take an active role in support of Chicago, where the Obamas lived before moving to the White House.
Chicago State University and the University of Illinois at Chicago are also interested in landing the library, as is Obama's native state of Hawaii.
Hawaii's bid is not considered serious, the two newspapers said, even though Obama was born and grew up in Hawaii and the Obamas spend Christmas vacation in the state.
Obama graduated from Columbia in 1983 with a bachelor's degree in political science after transferring to the university as a junior two years earlier.
He moved to Chicago in 1985 to become a community organizer, leaving for three years to earn a law degree from Harvard Law School in Cambridge, Mass., but then returning in 1991 to join the University of Chicago Law School.
For two years he was a visiting law and government fellow and for 12 years he taught constitutional law.
The Obama daughters attended the private University of Chicago Laboratory Schools before the Obama family moved to Washington.