Remain and Leave both have 35 per cent of support, with 30 per cent undecided, according to a TNS poll.
However, in previous polls a larger proportion of people, while not necessarily voting to remain, believed the outcome of the referendum would result in the UK's continued membership of the EU.
Belief in the UK remaining in the EU has fallen from 41 per cent to just 33 per cent, while the proportion of people who anticipate Brexit has remained at 28 per cent. Those who don't know have increased from 31 per cent to 38 per cent.
"A shift in the perceived outcome of the referendum could prove significant as the next ten weeks roll on. It shows that while the country remains decidedly split on how they will vote, people are beginning to sense a shift in the nation’s mood," Luke Taylor, head of social and political attitudes at TNS UK, said.
He added: "Although voting intention doesn’t show a winning side yet, this could be an indication of things to come."
The poll comes as campaigns attempt to woo voters ahead of the referendum on 23 June.