Davis, a Democrat, has posted several messages on her Twitter feed in recent days promising to announce "something big" Oct. 3, and multiple media reports indicate she will announce her candidacy for the Democratic nomination for governor in the 2014 election.
Davis has told supporters and potential donors she will officially announce her campaign next week, The New York Times reported.
"I can't wait to see you on October 3 when I announced what's next for me. RSVP today," she posted Friday.
"A week from today, I'm announcing something big," she posted Thursday. "Can you chip in now to show the strength of our grassroots network?"
Mark White, a Democrat who defeated incumbent Republican Gov. William Clements in 1982 and held office from 1983-87, told the Times Davis' chances of securing her party's nomination "are very good."
"I compare it to my chances of winning when I decided to run for governor," White said. "Everybody said it was impossible to do, and I was able to do it, and they're probably telling her the same thing. She's got the advantage that David had over Goliath."
Davis' filibuster helped kill the bill in a special session this summer but Gov. Rick Perry called another special session and the bill was passed. Abortion-rights advocates promised they will challenge its constitutionality in court.