Faafoi's office had a different account of the meeting, saying it was not a Five Eyes Nations meeting, although it did include members from the five countries.
His office repeated the Government's existing promise to work alongside partners in Afghanistan to respond to the rapidly evolving humanitarian situation.
Faafoi's office said there haven't been any discussions about a separate refugee system for Afghan nationals in conjunction with other partner countries beyond the United Nations refugee programme. Newe Zealand had no plans to change its current intake.
New Zealand has received 848 Afghan refugees since 2014.
The Five Eyes in its strictest sense is an intelligence-sharing alliance, although it is often used to refer to cooperation among the five nations in other policy areas.
Recently, New Zealand has expressed unease at the creeping remit of the Five Eyes.
The UK is the second Five Eyes country to pledge to resettle more Afghan refugees. This week, Canada promised to settle more than 20,000 Afghans from groups it considers likely targets of the Taliban.
According to the United Nations' refugee agency, about 250,000 Afghans have been forced to flee their homes since late May, most of them women and children.
Per head of population, New Zealand has one of the lowest rates of refugee acceptance in the world even after the current Government increased the annual refugee quota by 500.
It accepts 1500 refugees a year - or about 0.3 refugees per 1000 people. This ranks New Zealand 95th in the world.
Similar-sized countries like Norway and Ireland accept 11.29 and 1.22 refugees per 1000 residents, ranking them 15th and 69th respectively.