The poll finds that the Remain campaign now has a 13-point lead with just one month until the referendum, according to the Telegraph.
The latest ORB poll puts Remain on 55% and Leave trailing on 42%, among people who definitely intend to vote.
Amongst all voters, the Remain campaign now has a 20-point lead, with 58% of voters saying they back the pro-EU campaign.
In an analysis of the data, Sir Lynton Crosby, who masterminded the Conservative Party's general election victory last year, warns that the Leave campaign is "dwindling" and has "failed to quell ongoing concerns about the financial and economic consequences of a Brexit."
Read alsoDavid Cameron: ISIS, Vladimir Putin 'might be happy' with BrexitThe poll finds that men, Tory voters and over-65s are increasingly turning to the pro-EU campaign after previously supporting a Brexit in much larger numbers.
It is understood that the figures in the poll for the Telegraph reflect the private polling being done by Downing Street ahead of the June 23 vote.
In March, the polling found that 60% of Conservatives were intending to vote Leave compared to 34% for Remain.
Read alsoUK referendum: 51% back Remain campaignTuesday's poll finds that among Tory voters, 57% now say they will vote to stay in the EU, compared to 40% backing Leave.
The survey finds that 52% of over-65s now support staying in the EU, with 44% backing Leave.
