While Mr Burns insisted the prime minister has “a very keen interest”, he said he is not sure what an intervention from Mr Johnson would be “seeking to achieve”.
He went on to point out that the Assembly continues to sit and pass legislation, and that the Executive would have been dissolved within weeks for an election, regardless of the DUP’s actions.
Mr Paisley, speaking at the Northern Ireland Affairs Committee at Westminster, asked Mr Burns: “How do you account for the silence of the prime minister?”
The NIO minister replied: “The prime minister had been heavily involved. He has spoken to the secretary of state. The prime minister and I spoke on Friday evening when I was in Portaferry.
“The prime minister takes a genuinely deep interest in this.”
He continued: “The prime minister knows my interest in Northern Ireland and my background in Northern Ireland.
“It’s not exactly a state secret that the prime minister and I go back a bit, would consider each other friends and, certainly, strong allies.
“I would hope that the fact that he asked me, one of his close friends in the House of Commons, to serve in the Northern Ireland Office you could take as a declaration of his interest and commitment to Northern Ireland.”
Mr Burns added: “I don’t want to diminish what is going on but I’m not sure at the moment what a prime ministerial intervention would be seeking to achieve.
“We have been very clear that we would like your party to put the first minister back in.
“As I have said, and you will all know better than I, ministers are still in place, business is still being conducted, the Assembly is still sitting, legislation can still pass.
“The only things that cannot happen are those new initiatives that would require the sign-off of the Executive.
“We don’t want to over-dramatise what is going on. We are quite close to the period of dissolution anyway for a scheduled election which is only weeks away.
“But I can assure you that the prime minister is taking a very keen interest in this, and he is overwhelmingly committed to resolving the thing that Sir Jeffrey Donaldson has said is the reason for Paul Givan’s removal as first minister – the protocol.”
Responding, Mr Paisley accused the prime minister of a “calculated tactical decision to keep quiet” over Northern Ireland.
He stressed his belief to Mr Burns that it was a mistake for Mr Johnson not to speak publicly following the resignation of his DUP colleague Mr Givan last Thursday.
Mr Burns also insisted Mr Givan’s resignation “will not fundamentally alter” the UK’s negotiations with the EU over the protocol.
“I have to say very candidly that the government is clear on our intentions with the (European) Commission on the protocol, and the withdrawal of the first minister will not fundamentally alter the government’s determination to carry on engaging with the commission to find resolution, to find solutions to the situation in Northern Ireland, recognising the uniqueness of the position of Northern Ireland as part of the United Kingdom single market but also buttressing against the single market of the European Union in the form of the Irish Republic,” he said during the evidence session at the Westminster committee meeting.
“From a UK government point of view, in terms of protocol conversations, the plan is clear and the plan is continuing to be played out led by the foreign secretary.”
Mr Burns said the government “profoundly regrets” the withdrawal of the first minister and has “urged the DUP to get the first minister back in”.
“We have not actually witnessed a collapse of the institutions in the way they had before,” he said, adding that he noted a “degree of positioning” ahead of the Assembly elections.
Mr Burns also said he believed there would be a “very different reaction” from parties if an election was years away.