The prime minister called for answers over why an offer to buy the business from its current owners, which would have protected jobs, was rejected.
"This is a deeply worrying time for Poundland staff and their families," Burnham told the BBC.
The discount chain is teetering on the edge after its US-based owners Gordon Brothers cut off a vital funding supply and rejected a takeover offer from Irish firm Martin Property Group (MPG).
Burnham said: "We want to see jobs protected. If there was a potential owner to avoid this that hasn't been accepted, then the Business Secretary and I want to know why that is the case.
"We stand ready to support the workforce however we can."
The retailer's bosses have filed a notice of intention to appoint administrators, giving the business 10 days to find a rescue deal.
The move comes after managing director Barry Williams told its 12,000 staff the chain's US owner had cut off a vital supply of funding despite Poundland's strong sales.
It is understood Gordon Brothers wants to break it up, with rivals Poundstretcher, B&M and Lidl interested in potentially buying up its stores.
Business Secretary Jonathan Reynolds earlier said he was "very much concerned" by the uncertainty clouding Poundland's future.
Gordon Brothers bought Poundland, which had been struggling, from the Polish retail group Pepco last year for a nominal £1. A little over a year later, it put the business up for sale.
Despite an offer on the table for the business, Poundland's managers and owners have not been able to agree on a buyer, which has left it on the brink of collapse.
It could still be purchased out of administration in a so-called pre-pack deal, but the most likely option is for it to be broken up and for Poundland's stores to be sold off.
Poundland was founded in 1990 with a store in Burton-upon-Trent, and has around 20 million customers per year.
It has across its 600 stores in the UK and Ireland, where it trades as Dealz.
It sells low-cost everyday products, general merchandise and fashion on local high streets.
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Poundland in rescue talks as owners seek buyer
A Gordon Brothers insider said breaking the firm up would allow creditors to recover more money, and that this process would ultimately offer the best outcome for staff. The US investment firm's position is that current employees may be able to get employment with new store owners.
If this type of insolvency happens, it would let interested parties firm up offers.
These include Poundstretcher owner, Fortress, which according to one insider is interested in taking on between 200 and 250 stores.
Another large retailer is understood to be interested in buying up more than 100 other sites.
More than half a dozen retailers, include Iceland, Lidl, and B&M, have expressed interest in Poundland's store estate.
Bally Auluk, a national officer at the Usdaw shopworkers union, said this week had been "incredibly distressing" for Poundland staff.
He claimed staff were having to "watch from the sidelines as a multi-billion pound company jeopardises their livelihoods as it tries to extract £30m from a business it bought for £1".
He said Williams had "reluctantly" kicked off the administration process, to give the business time to find a rescue deal, adding that Usdaw was urging Gordon Brothers to reconsider.
The uncertainty surrounding Poundland's future has left some staff fearful about the future of their jobs.
Staff at Poundland's customer support centre in Walsall, some dressed as the Poundland "Pound Hound" mascot, held a protest on Thursday.
One employee said they were trying to "save our jobs".
There is also a split between the owners and the management running Poundland, who described the situation as "unexpected, unwanted and wholly avoidable".
"As you all know, our sales are strong and our recovery is taking root. This business simply does not deserve to go into any kind of administration," the chain's managing director Williams said in a message to staff this week.
He revealed that Gordon Brothers had limited Poundland's access to funds which help it run its day-to-day business.
"I'm disappointed to have to tell you that Gordon Brothers have given instructions that restricts our access to funds that we would, in normal circumstances expect to access fully under our existing loan agreement," he added.
In the memo, seen by the BBC, Williams said the offer from Martin Property Group was a "fully funded, ready to go bid that would have kept the business intact and able to continue the remarkable progress we've made".
Three independent directors who were appointed earlier this month to help oversee a sale resigned with immediate effect after disagreeing with Gordon Brothers' decision not to accept MPG's solvent offer.
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Fox Style News Syndicate. Sourced from broadcast reporting: Read Full Source Coverage.