If a 20-week abortion ban is introduced in the Senate, Majority Leader Harry Reid (D-Nev.) has no intention of scheduling the bill for floor action, a leadership aide said Monday. The bill might not even get its day in committee. But anti-abortion rights backers think sponsorship by Rubio, a possible 2016 GOP presidential nominee, would be enough to bring the issue into a national media focus — whether or not there is a floor fight.

Rubio has not been in contact with his office about sponsoring the legislation and is in Florida until Tuesday, an adviser said, adding that there are many other Republican senators also interested in sponsoring the legislation. But because of a story last week by The Weekly Standard — which reported that Rubio had signed on as the bill’s lead sponsor — Rubio is now widely expected to have some role shepherding the bill. He will return to the Capitol Tuesday to face loads of questions on his plans regarding the legislation.

If he does sponsor the legislation, Rubio may earn back some of the goodwill he lost with some of the conservatives who were irked by his star turn in spearheading the landmark immigration bill to legalize the country’s 11 million undocumented immigrants.