Alex Neil told a newspaper he personally believes the current limit should be looked at, although he would not be drawn on specifying a limit.
His comments came after UK Health Secretary Jeremy Hunt said he would like to see a ban on terminations more than 12 weeks into a pregnancy.
The intervention from the newly-appointed Scottish Health Secretary drew criticism from Green co-convener Patrick Harvie and women's rights campaigners.
Speaking to the Scotland on Sunday, Mr Neil suggested the cut proposed by Mr Hunt was not realistic but said he favoured a look at the current situation.
He said: "I do think there is a case to be had for a reduction from 24 weeks, but I don't know if 12 weeks is realistic, frankly.
"But I do think there is now a case, given the state of medical science and the fact that babies do survive from a much earlier stage in the pregnancy.
"I do think there is a case for looking to bring down the number of weeks, but that is a personal opinion."
Mr Neil said he would not pick a number weeks "out of the air", saying it was right to look at evidence on the subject.
Mr Harvie said Mr Neil's comments were "irresponsible".
He told the BBC's Sunday Politics Scotland: "As soon as I read that article I immediately put down a written question asking whether the Scottish Government remains committed to reproductive health services which are accessible and which support women's rights.
"But I think a new Health Secretary, so soon after being appointed, really should have been a bit more thoughtful about how his comments would be interpreted.
"I think this is an irresponsible way to present a personal opinion, so soon after becoming the new Health Secretary."
Following Mr Neil's comments, Kate Smurthwaite, of the Abortion Rights campaign group, said: "He seems to be saying that there's a case based on new medical evidence. I find that really surprising because there isn't any new evidence.
"The Royal College of Obstetricians and Gynaecologists published a report quite recently saying they saw no reason to change the term limit and they don't believe there's been any significant improvement in the survival or viability rates for pre-term babies.
"Personally I think that the notion we should build a law based around the survival rates of pre-term babies is a weird way of doing it. It's a red herring. It should be about adult women and their right to make choices about their own lives and their own bodies.
"I think it's so strange that politicians wade into this debate and feel that they have to start deciding what reasons are bad reasons (for having an abortion). For me that infantilises women, it implies that we're childlike and can't make these choices for ourselves.
"This is about our bodies. We have to accept that women are not stupid, women are intelligent beings and that they have the ability and the right to make these choices for themselves."
After the newspaper article emerged, Mr Neil issued a statement setting out the Scottish Government's current position.
He said: "There is no government or party policy on the issue of abortion - either now or in an independent Scotland - because it always has been and always will be a matter for the conscience of individual parliamentarians.
"Therefore, by definition, there are no government proposals to reduce the current 24-week time limit.
"This was a mischievous presentation of my comments on what everyone should remember is a serious and sensitive issue of conscience."
Mr Neil's comments came after the Prime Minister said the Westminster Government has no plans to put forward legislation to reduce the time limit for abortion.
David Cameron did, however, make it clear that there were opportunities for MPs to bring about a vote in the Commons, which would be treated as a matter of conscience.
Mr Cameron said that Mr Hunt was expressing his personal view, and not the position of the Government.
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