"There are a lot of people who call themselves Tea Party people that did like the independent mindedness of Herman Cain. So I'm optimistic that we'll pick up some votes from there," he said Sunday on CNN's "State of the Union".
But once high-flying contenders such as Perry and Rep. Michele Bachmann of Minnesota have not managed to bounce back so far, despite weeks of trying.
Bachmann said Sunday she was the "consistent conservative" in the race and her campaign would benefit most from Cain's departure.
"A lot of Herman Cain supporters have been calling our office and they've been coming over to our side," she said, also on CNN. "They saw Herman Cain as an outsider and I think they see that my voice would be the one that would be most reflective of his."
Cain's once-prospering campaign was undone by numerous allegations of sexual wrongdoing.
Gingrich, twice divorced and now married to a woman with whom he had an extramarital affair, has been the most obvious beneficiary of Cain's precipitous slide.
But Perry, Bachmann and possibly others are likely to make a play for Cain's anti-establishment tea party backing. Time is running short for them to establish themselves as the top alternative to Romney, who has long been viewed with suspicion by many conservatives.
Fouhy reported from New York.
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