Former Alaska Gov. Sarah Palin (R) remains unpopular with the general public and could face obstacles from voters in her own party should she seek the Republican nomination for president in 2012, a poll released Tuesday shows.

Sixty-four percent of the public says it would not vote for Palin if she decides to run. Among those who lean Republican, 41 percent say they would definitely not vote for her, up from 36 percent in December 2010, according to a Washington Post/ABC News poll.

In addition, 63 percent says she is unqualified to serve as president. In a hypothetical general-election match-up, President Obama leads Palin 55 percent to 40 percent among registered voters, his highest margin among six candidates polled. 

The poll is the first national survey to test Palin — the 2008 GOP vice presidential nominee — since she embarked on an East Coast bus tour on May 29, a move that provoked speculation she could jump into the GOP field.

Palin has jumped to second in the GOP primary field. She trails the front-runner, former Massachusetts Gov. Mitt Romney (R), 21 percent to 17 percent. Palin was polling at 5 percent support in April.

The poll suggests Palin's entry into the race could set up a two-way primary between her and Romney, with the rest of the field fighting for scraps. None of the other 10 candidates or potential candidates polled received double-digit support.

Still, other metrics suggest Palin would have a tough time winning the nomination.

Thirty-eight percent of those who lean Republican say they definitely would not vote for her for the GOP nomination, the highest out of a dozen candidates and potential candidates surveyed. Only 18 percent say they would definitely vote against giving Romney the nomination.

Twenty-four percent of those leaning Republican say they would strongly consider voting for Palin for the nomination, a figure equal to Romney's support.

The poll is based on telephone interviews with 1,002 adults nationwide between June 2 and 5. The full sample has a margin of error of 3.5 percentage points.