Speculation surrounding Rubio's role in the bill relates in part to his potential as a GOP presidential candidate.
Leading an effort against abortion rights would help Rubio with the Republican base, which has sharply criticized the immigration overhaul that he helped pass through the Senate this summer.
Pro-abortion rights political action committee EMILY's List fired a warning shot against Rubio Wednesday night after he expressed interest in sponsoring the abortion measure.
"Rubio continues to jockey for position as the most anti-woman, anti-family candidate of the potential GOP 2016 field," a spokeswoman for EMILY's List wrote in an email to reporters.
"He just can't help himself from pushing for dangerous legislation that echoes the nationwide assault on women's rights and freedoms. We will be watching, as will women across the country."
Once it is introduced, a national 20-week abortion ban is unlikely to see a normal floor vote in the Senate, which is controlled by Democrats.
But aggressive tactics by the GOP could force the issue — conservative senators could choose to filibuster a must-pass bill until an abortion vote is held, for example.
Debate over late-term procedures sprang to life during the trial of Kermit Gosnell, a Philadelphia abortion doctor who was convicted this spring of killing three infants born alive after failed abortions.
Gosnell was also found guilty of involuntary manslaughter in the death of an adult woman patient.
Abortion-rights opponents used the case to argue for restrictions on abortion late in pregnancy and for additional regulation of abortion clinics.
The GOP-led House responded to this call in June by passing its own 20-week ban based on the disputed premise that fetuses can feel pain at that stage of development.
Supporters of abortion rights argued that Gosnell was an anomaly among abortion providers, and that restrictions on abortion rights, such as waiting periods and parental consent laws create business for illicit, unsafe clinics.
Pro-abortion rights groups also say that 20-week bans are unconstitutional based on established Supreme Court precedent.
Rubio's comments were first reported by Politico.
— This story was updated at 11:44 a.m.