The country is also said to be in talks with the EU, US and Japan.
The first round of trade talks between the UK, Australia and New Zealand is expected to begin via video conference in the coming weeks.
"Ambitious, wide-ranging free trade agreements with old friends like Australia and New Zealand are a powerful way for us to do that and make good on the promise of Brexit," UK International Trade Secretary Liz Truss said.
Johnson did not single out any New Zealand biscuits that he'd like to get at more reasonable prices but put out a separate video announcing the trade talks with the country and adding that the UK already buys "colossal quantities of New Zealand wine".
He points out that the UK exports gin and buses to New Zealand.
"I'm proud to call the United Kingdom a friend of New Zealand, and am thrilled that we now have the opportunity to strengthen our alliance by negotiating a new free trade agreement," he wrote in his tweet.