Labour claimed the Nationalists, who started the campaign saying independence would be one of their "three priorities", were now avoiding the subject because it had gone down badly on the doorstep.
It came as even an SNP MSP predicted people in Dunfermline would reject independence in 2014.
Writing on his Facebook page last week, Glasgow Shettleston MSP John Mason said that after campaigning in Dunfermline: "It was clear to me that most people want independence but plan to vote No because they are nervous about it."
With both parties mounting a final push ahead of Thursday's ballot, SNP strategists conceded Labour were ahead, and said the Nationalists had to be "realistic" about their chances of a win.
"It's still Labour's to lose but we are working hard to win," said one.
"It's the mid-term of a second term of an SNP government. We have definitely got the best candidate. But we have got to be realistic."
SNP sources said Labour had identified their supporters early and pushed hard on postal voting, while the SNP had suffered from a lack of voter information in a seat they won from third place in 2011.
"We've been playing catch-up," said a source.
Caused by the resignation of disgraced former SNP MSP Bill Walker, who was jailed for a year last month for 23 counts of domestic abuse, the by-election has been seen as Labour's to lose.
Besides the circumstances of Walker's exit, his majority in 2011 was just 590, meaning Labour need only a 1% swing to win amid a mid-term decline in SNP support.
Professor John Curtice of Strathclyde University said it would be "staggering" if Labour candidate Cara Hilton didn't win.
Launching the campaign with SNP candidate Shirley-Anne Somerville a fortnight ago, Alex Salmond said the party would campaign on local school closures, protecting public services, and an "optimistic, upbeat campaign for a better future through an independent Scotland".
But while it has dominated this weekend's SNP conference, independence appears on the backburner in Dunfermline. Labour, the LibDems and Tories all reported voters raising independence as a negative on the doorstep.
A Labour spokesman said: "It's no surprise that the SNP have dropped independence from their campaign - the people of Dunfermline are overwhelmingly against their plan to break up Britain. It must be particularly galling for Shirley-Anne Somerville who has spent the past year working on the pro-independence campaign [as communities director of Yes Scotland].
"Her political dream has been ditched and instead the SNP have turned this campaign into one of the most negative in living memory."
If Labour win it would be the first time a seat has changed control in a Holyrood by-election since 2000, and would be seen as an ill omen for the independence vote.
If Labour failed to win it would be a shock to party morale and damaging to Scottish leader Johann Lamont personally, as the SNP's other main attack has been against Lamont's review of universal services.
A Labour source said the party knew where its support lay, and would be trying to maximise turnout on Thursday, but added that, despite predictions of a comfortable win, the seat was still "in play".
Somerville said: "Labour are reduced to making up claims about the SNP campaign that are simply untrue."