Australian Prime Minister Malcolm Turnbull has jetted out of New York confident that ties with the US and President Donald Trump are rock solid.

Just before his departure for the airport Mr Turnbull repeatedly used one word to describe the Australian-US relationship: family.

That could seem apt to describe what has been a rollercoaster ride for Mr Turnbull and Mr Trump since the president took office on January 20.

Family members are known for having disagreements and then making up.

Mr Turnbull, his wife Lucy, Mr Trump and First Lady Melania spent more than three hours together behind closed doors and in full view at Thursday night’s event to honour the 75th anniversary of the Battle of the Coral Sea.

“It was great for Lucy and I to meet with the president and Mrs Trump,” Mr Turnbull told reporters at the Morgan Stanley offices near Times Square.

“Again, that was more family than formal.”

Australian and US officials were breathing sighs of relief after the two leaders seemed to enjoy each other’s company.

Mr Trump, in his haphazard way, denied his January 28 phone call with Mr Turnbull was a rocky one.

He first told reporters at the Battle of the Coral Sea event the call was great, and Mr Turnbull agreed.
 But later in the night Mr Trump admitted the call was “testy”.

Immigration discussed

Mr Turnbull would only offer limited details about his bilateral meeting with Mr Trump, but said they did talk about immigration.

Mr Trump came to power vowing to halt illegal immigrants entering the US and forcing refugees to undergo extreme vetting.

Australia has had success in stemming the flow of refugees, but Mr Turnbull, asked if it was discussed with Mr Trump, said “we don’t ever presume to advise or counsel other countries on how they should manage their affairs”.

Mr Turnbull wrapped his two-day visit with a breakfast meeting with 20 chief executives including News Corp’s Robert Thomson, Blackstone’s Stephen Schwarzman, Alcoa chief executive Roy Harvey and Morgan Stanley chief executive chairman James Gorman.

Golfer Greg Norman, chairman and chief executive of Shark Industries, and Bloomberg founder Michael Bloomberg also attended.

—AAP