A new set of polling analysis revealed at the Labour conference has shown that a small swing from the SNP would see Labour win as many as 28 seats in the next general election and help Jeremy Corbyn become prime minister.

The analysis, from polling company Survation, showed that there are 21 seats Labour could win from the SNP with a swing of less than 5%, on top of the seven it won in June's general election, and recent polling has shown that SNP voters are willing to be won over to Labour with the right policies.

At the Survation event at the Labour conference in Brighton on Monday, Scottish Labour general election candidate and Labour national executive committee member Rhea Wolfson also said the party must abandon its "incredibly negative" message to take advantage of its recent resurgence.

Damian Lyons Lowe, the chief executive of Survation, told the event that Scotland could be a "really, really big factor" in delivering a majority government for Labour in future general elections. He did warn, however, that the situation is Scotland is fragile and just a 3% swing back to the SNP could see Labour reduced to one seat.

"There are 21 seats that require less than a 5% swing for Labour to win and every single target is an SNP-held seat, so the situation in Scotland for Labour is on a knife edge," said Lyons Lowe. "A 3% swing against Labour to SNP would take Labour back to just the one seat held in 2015 – Edinburgh South. However, just a 5% swing from the SNP would give Labour 28 seats in Scotland."