Perry also faces opposition to one of his signature immigration policies in Texas, the survey shows.
His rapidly changing fortunes underscore the fluidity of the Republican race and the lingering dissatisfaction with the candidates.
That has led some major donors and party leaders to urge New Jersey Gov. Chris Christie to declare his candidacy.
Christie is feverishly assessing whether to do so, with a decision expected this week. But the Post-ABC poll finds only modest public support for a Christie candidacy. About 42 percent of Republicans and GOP-leaning independents say they would like to see the New Jersey governor join the race. Thirty-four percent say no, with the rest offering no opinion.
That finding is far more positive than the receptivity to a candidacy by Sarah Palin. Two-thirds of Republicans say they do not want the former Alaska governor to seek the party’s nomination.
Although not fully satisfied with their choices, Republicans are optimistic about their chances of winning the election. More than eight in 10 say the eventual GOP nominee is likely to beat President Obama next year. In the new poll, Obama’s approval remained at a low point in his presidency.
Among announced candidates — without Christie or Palin in the race — Romney leads with 25 percent, which is identical to his support from a month ago. Perry and Cain are tied for second with 16 percent, numbers representing a 13-point drop for Perry and a 12-point rise for Cain since early September.
Rep. Ron Paul (Tex.) is the only other candidate in double figures, at 11 percent. Just behind him are former House speaker Newt Gingrich (Ga.) and Rep. Michele Bachmann (Minn.), both with 7 percent. Gingrich’s support has held steady through the late summer. Bachmann’s numbers fell sharply after Perry announced his candidacy.
Former senator Rick Santorum (Pa.) and former Utah governor Jon Huntsman Jr. bring up the rear, with Santorum at 2 percent and Huntsman at 1 percent.
If Christie and Palin are included in the hypothetical matchup, he checks in at 10 percent and she at 9 percent.
Perry’s support for the Texas policy of providing in-state tuition to the children of illegal immigrants appears to be a significant problem in the GOP race. About two-thirds of Republicans and GOP-leaning independents say they are less likely to vote for a candidate who backs such a policy. Among tea party supporters, nearly eight in 10 say this position is a negative factor.