While 75% of Palestinians backed the contents of the 'prisoners document', only 47% said they would vote in favour of it in a referendum if it were held today.
The survey, carried out by the Palestinian Center for Policy and Survey Research, said 44% would vote against the document.
The document, drawn up by Palestinian prisoners in an Israeli jail, implicity acknowleges the existance of Israel and the supremacy of the Palestinian umbrella organisation, the PLO.
Hamas is opposed to key elements of the proposal.
Abbas has said he will hold a referendum on the document on July 26 unless Hamas agrees to support it.
"The sympathy, when it comes to the referendum, is with the underdog, and in this case that is Hamas," said Khalil Shikaki, the director of the PCPSR.
"Most people like the document as it stands, but the climate of opinion is not in favour of a referendum," he added.
However, there are indications that Hamas and Abbas's Fatah movement may agree on a reworked document in the days ahead, which could cancel the need for a referendum and ease tensions.
Faction members convened in Gaza City on Monday to try to hammer out a deal.
Around 20 people have been killed in fighting between Hamas and Fatah in the past month in the Gaza Strip.
The results of the survey, which polled 1,270 adults between June 15-18, are likely to encourage moderation by both sides, Shikaki said, with Abbas edging away from a referendum showdown and Hamas cautiously supporting a popular document.
"We might finally be seeing the debate giving way to the more moderate wing of Hamas," he added.