The source said trade secretary Liz Truss would be driving a hard bargain on key areas of interest.
They said: "New Zealand will need to give us more on services, mobility and investment if they want a deal.
"If we have to go beyond then into September to get the best deal, then so be it."
Any deal is likely to take a similar shape to the tie-up with Australia, which proposes a widespread liberalisation including the removals of tariff quotas on agricultural exports and has drawn criticism from British farmers.
It is expected to have a negligible effect on UK GDP, with modelling by the trade department even indicating that an extensive deal could mildly reduce Britain's national output.
Recent events have provided a potentially awkward backdrop for what may be the final weeks of talks.
Streaming giant Amazon last week announced that it would shift production of its highly-anticipated Lord of the Rings TV series from New Zealand to the UK, in a major blow to the country's creative and tourism industries.