Downing Street aides are planning to lobby Andy Burnham during access talks to revive the idea of “war bonds” to pay for higher defence spending when he becomes prime minister, the Guardian understands.

Senior No 10 figures want the Treasury to be allowed to borrow more for military spending and will try to convince Burnham to invest beyond the £13.5bn earmarked for the long-awaited Defence Investment Plan (Dip).

Defence aides are also understood to have travelled to Makerfield during the byelection campaign to bring Burnham’s team up to speed on the state of the UK’s depleted defence capabilities.

Although Burnham is now expected to enter Downing Street with the Dip already signed off, he will still come under heavy pressure from military leaders and others within the government to increase defence spending in the long term.

Keir Starmer confirmed on Wednesday he would announce the Dip before the Nato summit in Ankara on 7 and 8 July even though he leaves the following week, with No 10 arguing that while it is a big spending commitment, it is an existing one.

But he has faced criticism from some Labour MPs for pressing ahead with such an important policy issue, rather than leaving it to his likely successor who has told them he wants to give the Ministry of Defence (MoD) more than the £13.5bn on offer.

Burnham’s allies say that if the row over defence spending has been resolved by the time he takes over – amid signs that a little extra money has been found and the settlement accepted by military leaders – then he would probably just move on.

However, his team is reserving the right to reopen the Dip if that is not the case – with, for example, mismanaged programmes such as tanks investment left unaddressed and the concerns of the defence establishment unallayed.

Burnham is believed to have held discussions with John Healey at the the likely next prime minister’s request, who pushed Starmer’s premiership to the brink when he quit as defence secretary earlier this week, arguing the Dip fell well short of what was required.

On Wednesday the head of the armed forces, air chief marshal Sir Richard Knighton, said that Britain needs to fund capable armed forces that ​can go “toe to toe with Russia” and beyond, because if deterrence ​were to fail​ ​an all out war would be more costly.

Emphasising the need for investment, Knighton said that the UK needed “armed forces that our adversaries are scared of” at a conference organised by the Royal United Services Institute.

British defence spending “went from 2.9% in 1936, to 9% in 1939, to 52% of GDP in 1945​,” Knighton said, dramatically beyond the target spend of 2.7% of GDP next year, even though few believe all out war is likely.

With Starmer giving the green light to access talks, No 10 sources said they would use them to encourage Burnham to reconsider defence bonds, which have consistently been blocked by the Treasury amid warnings over the impact of higher borrowing.

The government had come close to issuing war bonds to pay for higher defence spending before. The Guardian understands that Morgan McSweeney, Starmer’s former chief of staff, had started pushing for them before he quit. But the plan was rejected by the Treasury.

Under the plans drawn up in No 10 and championed by Starmer’s business adviser Varun Chandra, the government would have issued up to £20bn of bonds which would have been exempt from inheritance tax.

The money raised would only be used to pay for defence spending – a limitation the Treasury has always resisted in the past. Because there would be a tax benefit to owning them, advocates of the idea believe the government would have had to pay a lower interest rate, helping bring down its overall cost of borrowing.

They also say it would have encouraged domestic individual investors to buy bonds, something that has faded in recent years as large hedge funds have become an increasingly important source of credit.

Some in government believe the UK would be paying less for public borrowing if more domestic investors owned bonds, as they believe hedge funds are more likely to sell gilts because of small fluctuations in the market.

Other Labour MPs have pressed for the UK to back the Defence Security and Resilience Bank initiative, led by Canada. Countries are being asked to sign up at the Nato summit next month, but the scheme supported by Healey when he was in government was opposed by the Treasury.

Alex Baker, a member of the defence select committee, said in a recent article the bank, which would specialise in lending to small defence companies would “turn political commitments into industrial output” and that it would “create the conditions for more factories, jobs, exports, innovation and resilient supply chains”.

Proponents argue that for a UK subscription of £900m, the UK would gain access to a bank with €100bn (£86bn) of lending capability. Earlier this week, Rachel Reeves, the chancellor, said she was talking to Canada – but also developing a “multilateral defence mechanism” of off balance sheet financing with Finland and the Netherlands.

During a cabinet meeting on Tuesday, Starmer told his ministers he wanted to “resolve difficult issues” in his remaining weeks in office to help with the transition, including publishing the Dip which has been delayed over funding disputes between No 10, the Treasury and the MoD.

Dan Jarvis, the new defence secretary, has signalled he is already looking for more funding and said he was involved in “very good and constructive” discussions. Other departments have been asked to make cuts to their budgets to allow Starmer to reallocate funds.