Rubio hasn't contacted his office about sponsoring the legislation and was in his home state through Tuesday, the adviser told Politico.
However, a July 2 story in the The Weekly Standard reported Rubio "agreed to be the lead sponsor of a Senate bill to ban abortion after an unborn child is 20 weeks old." The Weekly Standard article did not include any comments from Rubio or his office.
When the adviser eventually talked to The Weekly Standard, the adviser "couldn't confirm" the story, Politico reported Monday.
"When I did talk to them, I told them the same thing I told everyone: '[Pro-life groups] have been asking since April,'" the adviser told Politico Monday.
A similar nationwide ban passed the Republican-led House.
Even if such an abortion bill is introduced, a leadership aide said Senate Majority Leader Harry Reid, D-Nev., doesn't intend to schedule the measure for floor action, Politico said. A committee hearing isn't likely, either.
But abortion opponents said they think having Rubio's name attached to such legislation would be enough to focus national attention on the abortion controversy.
Rubio, who wowed the Republican National Convention in 2012, has been mentioned as a 2016 GOP presidential nominee.
If he does sponsor the abortion legislation, Rubio could earn back some of the goodwill he lost among conservatives angered by his role in the sweeping bipartisan immigration legislation that would, among other things, provide a path to citizenship for the 11 million undocumented immigrants in the country, Politico said.