Immigration Minister Peter Dutton’s explicit linking of the arrangements to send Australia’s offshore refugees to the US and to accept some from Costa Rica presents not just an immediate credibility problem for Malcolm Turnbull but, potentially, a more serious longer-term one.

It contradicts the prime minister’s flat – if unconvincing – denial of such a link. It also raises the question, why would people believe Turnbull on anything remotely related to this issue in the future?

And that could be important if the Trump administration were to ask Australia to boost its military commitment in the Middle East.

Turnbull says any such request would be considered on its merits.

If there was a request and Australia were to agree, he would deny that the acquiescence had anything to do with his managing to twist Donald Trump’s arm to accept the deal Australia did with the Obama administration to take people from Nauru and Manus Island.

But that denial – always likely to be questioned – would be an even harder sell now.

In September, after the Costa Rica arrangement was announced, Turnbull was asked whether it had any material impact on the government’s ability to find homes for people on Nauru and Manus Island.

“It is not linked to any other resettlement discussions,” he said. “The announcement today is not connected to any other arrangements.”

This became the mantra, including after the deal about Nauru and Manus Island was announced following the presidential election. Dutton said on November 14: “The Costa Rica arrangement had nothing to do with this deal and it’s not a people swap.”

On Tuesday’s Bolt program on Sky, Dutton predicted the first offshore refugees would move in the next couple of months. Asked then when the first people from Costa Rica would arrive, Dutton said: “Well, we wouldn’t take anyone until we had assurances that people were going to go off Nauru and Manus … We want an outcome in relation to Nauru and Manus.”

“One of the lessons we’ve learnt from past arrangements, say the Malaysian deal for example that Julia Gillard entered into, we accepted all the people from Malaysia, not one person went from Australia. So we’re not going to be sucked into that sort of silly outcome.”

It should be said this is more than a bit rich. The people didn’t go because the Coalition opposition blocked the “swap”.

Bolt pressed Dutton on the arrangements with the US. “So it was a deal? It was, we’ll take yours if you take ours.”

Dutton said it wasn’t a “people-swap deal” but added: “I don’t have any problem with that characterisation if people want to put that”.

It’s always defied common sense to think there was no link between the Costa Rica and Nauru/Manus Island deals, and the government was taking the public for mugs to try to argue that. Now it is paying the price.

It remains unclear what the Americans honouring the deal will amount to, given it is up to them how many of the people they finally accept after Trump’s “extreme vetting” process.

Dutton’s proposition that the refugees from Costa Rica can’t come until he’s sure some of the offshore people are going suggests he feels the need to take out insurance.

Fairfax’s Michael Gordon has suggested Dutton could have handed Trump an excuse to junk the Manus/Nauru deal if he was so minded.

Foreign Minister Julie Bishop, in Washington for wide-ranging talks with the Trump administration, said on Wednesday: “The agreement is progressing and our officials are working together with United States officials to vet the applicants for settlement in the United States.” She wouldn’t be drawn on detail.

Asked whether she would characterise it as a swap deal, Bishop said: “That’s not the way I would categorise it.”

The government continues to fall victim of its own spin.