US President Donald Trump will honour the asylum-seeker deal with Australia "in some way", White House spokesman Sean Spicer has said.

Mr Spicer told press Australian Ambassador to the US Joe Hockey's meeting at the White House with Mr Trump's chief of staff Reince Priebus and senior adviser Steve Bannon was "a productive and candid conversation".

The comment comes after an unusually volatile week between the two nations over the contentious refugee deal.

The White House has launched an investigation into how embarrassing details of Donald Trump's tense phone conversations with counterparts in Australia and Mexico were leaked to news organisations.

President Trump cut short a phone call with Malcolm Turnbull after an acrimonious discussion about a refugee swap deal, a conversation which threatened ties between the two allies after details appeared in The Washington Post.

In an earlier call with Mexican President Enrique Pena Nieto about paying for a wall on the southern US border, Trump said he might send the US military to Mexico to stop drug cartels, which was leaked to the Associated Press.