Speaking at a news conference alongside Prime Minister Boris Johnson in Berlin, she stressed it would be up to the UK to offer a workable plan.
The PM said he was “more than happy” with that “blistering timetable”.
He accepted the “onus” was on the UK, but said he believed there was “ample scope” for a new deal to be reached.
In his first overseas visit to a fellow leader, Mr Johnson is meeting Mrs Merkel after he told the EU the backstop – which aims to prevent a hard Irish border after Brexit – must be ditched if a no-deal exit was to be avoided.
He will meet French President Emmanuel Macron on Thursday, before attending the G7 summit on Saturday alongside other leaders including US President Donald Trump.
And – despite Mrs Merkel’s comments – that message was echoed by Mr Macron on Wednesday evening.
“Renegotiation of the terms currently proposed by the British is not an option that exists, and that has always been made clear by [EU] President Tusk,” he told reporters in Paris.
“The backstop has always been a fall-back option until this issue is solved,” she said.
“It was said we will probably find a solution in two years. But we could also find one in the next 30 days, why not?”
Mr Johnson replied: “You rightly say the onus is on us to produce those solutions, those ideas […] and that is what we want to do.
“You have set a very blistering timetable of 30 days – if I understood you correctly, I am more than happy with that,” he added.
The prime minister has insisted he wants the UK to leave the EU with a renegotiated withdrawal deal, but the UK must leave on 31 October “do or die”.
Mr Johnson called the backstop “anti-democratic” and “unviable”.
Boris Johnson will certainly be pleased the German chancellor has left a door open.
Boris Johnson says it’s his job to find a solution and accepted a deadline of 30 days to come up with one.
The pressure is firmly on the UK to find that solution – and it’s going to be a huge challenge to put it mildly.
SNP Westminster leader Ian Blackford has confirmed he would attend the meeting next Tuesday, but warned the Labour leader that “all options must be on the table”.
Mr Corbyn has proposed that in order to prevent a no-deal exit, opposition MPs should help him defeat the government in a no-confidence motion and install him as a caretaker PM.
If he wins the vote, he plans to delay Brexit, call a snap election and campaign for another referendum.
But the Liberal Democrats, and some potential Tory allies opposed to a no-deal exit, have indicated they won’t back a plan that leads to Mr Corbyn in No 10.