According to the survey - published on Saturday by The Guardian and conducted by the Messina Group among 5627 people between September 12 and 24 - 49 per cent said they would support the UK rejoining the EU, while 28.5 per cent opposed it.
The Guardian said the findings could strengthen the position of those in government who want Labour to consider committing at the next general election to negotiating the UK's return to the bloc.
At the same time, some Labour figures have urged caution over reopening divisions associated with Brexit.
Burnham recently said he would examine "all options" for the UK's relationship with the EU and did not rule out the possibility of "going all the way" to seeking membership again.
At the Labour Party conference in September in Liverpool, Burnham said Brexit, which was approved in a 2016 referendum and took effect in 2020, "did more harm than good".
The Guardian reported the survey found Labour could increase its vote share if it proposed a new referendum and campaigned for the UK to rejoin the EU.
However, some Labour MPs, particularly those representing constituencies that voted to leave the EU, have expressed concern that even discussing re-entry could carry political risks and strengthen Reform UK, according to the newspaper.
Labour's 2024 election manifesto ruled out joining the EU single market and customs union.
Burnham has since signalled the government is reviewing its options for the UK-EU relationship ahead of a planned summit with the bloc.
The possibility of rejoining the EU has support among pro-European figures and parties, including the Liberal Democrats and Scottish National Party, while the Conservative Party and Reform UK oppose it.
