A House of Commons briefing paper details the extent to which working-age migrants from different countries are receiving benefits from the UK welfare state. The quick answer is: to a very small extent. Migrants are less likely than British-born citizens to receive benefits in work, and much less likely to receive benefits out of work. There is nothing in the figures to support the contention that the benefits system is under pressure from migrants.
The Migration Observatory at Oxford has also recently published a report on the minimum income requirement for non-EEA nationals. They were asked to review at what level of income people wouldn’t be dependent on benefits: it’s not a meaningful question, because benefits don’t work that way. The report cites a finding that the UK already has the most restrictive policies of 38 high-income countries.