However, fewer people now than in 2011 think immigration is bad for the economy - 47% in 2013 compared with 52% two years previously, new findings from NatCen Social Research's British Social Attitudes survey found.
The latest results of the survey, to be revealed in a BBC Two documentary The Truth About Immigration, tonight at 9.30pm and presented by political editor Nick Robinson, come nearly a week after access restrictions to the UK labour market were lifted for Romanians and Bulgarians.
Penny Young, chief executive at NatCen Social Research, said: "British Social Attitudes shows that public desire for a cut in immigration to the UK had begun to rise even before the restrictions on migrants from Romania or Bulgaria were lifted at the start of the year.
"Moreover, a majority of people who think immigration is good, economically or culturally, for the UK still want to see it cut.
"These findings highlight the complexity of this issue for politicians facing two elections in 18 months and with limited options if they want to attempt to reduce migration from Europe.
"The public broadly agrees that immigration is too high, but there are stark social divisions over the economic and cultural benefits of immigration."
The research shows 54% of respondents see immigration as good for the economy, but 55% of those who see it is as culturally beneficial also want to see it reduced.
The survey also revealed divisions among political party supporters.
Some 40% of Labour party supporters think immigration is bad for the economy but 36% believe it is good for the economy, the research shows, while 40% think immigration is bad for British culture and 41% see it as good for British culture.
Some 52% of Conservatives believe Britain's cultural life is undermined by immigration compared with 20% of LibDems.