Robert F. Bauer, the chief counsel for the Obama campaign, made the demand Tuesday in a letter to the Federal Elections Commission, the newspaper said.
The complaint argues Rove's group, Crossroads GPS, is a "political committee" and therefore subject to federal political contribution reporting requirements. Crossroads GPS describes itself as a "social welfare" organization, exempt from the kinds of financial reporting requirements imposed on political organizations, including requirements to identify contributors publicly.
Citing a ruling by the 4th U.S. Circuit Court of Appeals -- Real Truth About Obama vs. FEC -- Bauer's complaint asserts the government has the responsibility of determining the "major purpose" of Crossroads GPS and similar organizations.
"Crossroads seems to believe that it can run out the clock and spend massive sums of money in this election without accounting for a trace of its funding," Bauer wrote in the complaint. "Now, a federal appellate court has issued a ruling that makes clear that Crossroads is out of time."
A spokesman for Crossroads GPS, Jonathan Collegio, said Obama's lawyers were singling out Republican-friendly groups, the Times reported.
Crossroads GPS and other right-leaning groups are raising vastly more money from large donors during the current election cycle than liberal groups have raised.
University of California, Irvine, election-law expert Rick Hasen told the Times the court of appeals ruling makes it clear Crossroads GPS and similar groups violate U.S. law by refusing to disclose donors.