(CNN) - Sen. Marco Rubio is considering a request by anti-abortion groups to sponsor a bill in the Senate that would ban most abortions after 20 weeks, an adviser to the Florida Republican confirmed to CNN.
"I can tell you that the pro-life groups are asking Senator Rubio to sponsor the bill in the Senate. He's on a family vacation this week and will decide when he returns to D.C. next week," said the adviser, who asked for anonymity to speak more freely.
The story was first reported by the Weekly Standard, a conservative opinion magazine and website. The Weekly Standard's story included several sources saying that Rubio had agreed to be the lead sponsor of the bill.
Marjorie Dannenfelser, president of the Susan B. Anthony List, said her organization's been leading lobbying efforts to recruit Rubio to sponsor the bill in the Senate. She told CNN, "He wants to and that's good news."
Susan B. Anthony List describes itself as a group "dedicated to electing candidates and pursuing polices that will reduce and ultimately end abortion."
Other anti-abortion groups are also urging Rubio to sponsor the bill.
"We think Rubio would be a wonderful chief sponsor, should he choose to," Douglas Johnson, legislative director of the National Right to Life Committee, told CNN.
If Rubio does sponsor the legislation, it would give the measure added media attention, as Rubio is one of the GOP's biggest stars and is considered a possible 2016 Republican presidential contender.
But the first-term senator has drawn criticism from conservatives the past couple of months over his big push for a bipartisan comprehensive immigration reform package that includes a pathway to citizenship for most of the 11 million undocumented immigrants in the country.
The Republican-controlled House passed a similar late-term abortion ban bill last month, with only six Republicans voting against the measure and only six Democrats voting in support of it. The bill prohibits most abortions for women beyond their 20th week of pregnancy.
While the original House bill included an exception for cases in which the health of the mother is in danger, Republicans got strong pushback for not including exceptions for pregnancies that resulted from rape or incest. Republican leaders later added new language to include those exceptions.
The measure is expected to go nowhere in the Democratic-controlled Senate and the White House has threatened a veto on the legislation.
Republican Sen. Mike Lee of Utah, who along with Rubio was elected in 2010, proposed similar legislation in May that would ban abortions after 20 weeks in the District of Columbia. It was referred to the Committee on Homeland Security and Governmental Affairs.
The federal push comes after a number of states have passed such bans in recent years, including Nebraska, Kansas, Louisiana, Oklahoma, Indiana and Alabama. Arkansas has a ban in place for pregnancies beyond 12 weeks, and North Dakota has the tightest restriction in the country at six weeks, when a fetal heartbeat can be detected.
Texas is among the latest states attempting to end abortions after 20 weeks.
Gov. Rick Perry called a special session to try and pass a controversial measure that would prohibit most abortions after 20 weeks of pregnancy. The bill was sidelined last week following a filibuster by a Democratic state senator that captured national attention. Early Wednesday morning a House committee passed the measure and sent it to the full House, which with the Senate is in recess until next week.
Forty-eight percent of Americans questioned in a United Technologies/National Journal poll conducted late last month said they supported a bill that would ban most abortions after 20-months, with 44% opposed. According to the national survey, by a 50%-44% margin, women supported the 20-week ban, with men divided 46%-45%. There was also an expected partisan divide, with 59% of Republicans and 53% of independents, but only 33% of Democrats supporting such a bill.
Meanwhile, North Carolina is also in the midst of a heated battle over abortion restrictions. The state Senate passed a second reading of a bill Tuesday and a final vote is expected Wednesday. The measure doesn't include a 20-week provision but would place increased standards on abortion clinics and limit health care coverage for abortions.
And a recent move by Ohio's Republican governor is also grabbing headlines. John Kasich on Sunday night signed into law the state's budget, which includes some controversial abortion provisions like requiring ultrasounds for anyone seeking an abortion and limiting clinics in which abortions are performed from transferring patients to public hospitals, if those patients need more medical care after a procedure. The budget will also make it more difficult for family planning groups in the Buckeye State to obtain funding for preventive care. And it puts Planned Parenthood behind clinics that don't provide abortions, when it comes to obtaining state funding.