"He has a more casual attitude to diplomatic relationships than is usual, but he conveyed his enthusiasm for meeting at some stage in the White House ... 'well, if you're passing by'."
Despite the warm offer, English said he was unlikely to get there before the September election. "I'd imagine it will take some time for the new administration to bed in, and then we've got the election campaign. I wouldn't anticipate getting there this side of the election."
He said Trump appeared to be positive about the relationship with New Zealand. "He thinks it's a fantastic place. He likes the idea we are a long way away so therefore we aren't under the same pressures as everybody else."
They talked about Trump's attempt to ban citizens from seven countries entering the US - an Executive Order which has been suspended by the US courts.
English told Trump he disagreed with the action and it was not something New Zealand had done. "He just noted our views. I don't think that he was surprised by people having a different view."
The pair also discussed the different ways they dealt with border security.
"The discussion focussed on what steps we take and the US takes to as a way of protecting our citizens from high-risk people coming in, which is clearly at the top of his agenda."
They also discussed trade - Trump has withdrawn the US from the TPP and is instead planning to try to negotiate bilateral agreements with other countries as part of his "America First" policy.
English said the topic of a New Zealand-US agreement did not come up.
"He's clearly focused on bigger agreements such as NAFTA. And in any case we want to sort through our own process to be sure that kind of a deal would be better than, say, a TPP proceeding without the US."
No request was made for New Zealand to do more to combat Isis.
"[There was] I think a good understanding that we discussed our role as a small country, a long way away, pulling our weight in the defence of our own people around the world and working alongside the US to contribute."
English and Trump also discussed New Zealand golfer Sir Bob Charles and the Superbowl - English said he had thanked Trump for taking time out to call on the day of the Superbowl.
"He knew a reasonable bit about New Zealand. He asked about the economy and is a great admirer of Bob Charles through his golfing contacts."
English had also passed on his thanks to Trump for the visit of the USS Sampson to Kaikoura in the days after the earthquake.
A short time ago the White House issued a readout of the call between the two leaders:
English said "at the top" of Trump's agenda appeared to be security and the safety of US citizens.
Trump used positive language and said he thought New Zealand was a fantastic place.
He said they spoke for about 15 minutes.
Trump reportedly described his phone call with Turnbull as the worst call he'd held that day.
Asked if he had ranked English's call, English laughed. "I'm sure he thinks it was a fantastic call. He certainly thinks very highly of New Zealand."
Judging from his Twitter account, Trump moved on fairly quickly - his only tweet after the call was about the Superbowl.
Prior to the call, English had said he would raise the travel ban the Trump executive order had imposed on nationals from seven countries, and the suspension of refugee processing. That order has currently been suspended by the courts in the US.
English expected the call to be "civil" despite the reports of a fractious phone call between Australia's Prime Minister Malcolm Turnbull and Trump.
That was over the deal struck between the Australian Government and the Obama administration for the US to take about 1200 refugees from Manus Island and Nauru.
The details of that call emerged a few days after it took place.
In a summary of Trump's 'second week of action', The White House said the aim of the calls are to "promote an America First foreign policy."
English is about the 15th leader Trump has spoken to since his inauguration.
Former Prime Minister John Key had a brief conversation with him after the US elections. He has spoken to leaders from Israel, Russia, Germany, Mexico, India, Japan, France, Saudi Arabia, United Arab Emirates, Korea, Jordan, China, Britain and Australia.
