But other guests on the television show were dismissive of the possibility of an internal party coup against the chancellor.
“Anyone who tries to overthrow some one like her will destroy himself,” Karl-Rudolf Korte, a political scientist at the University of Duisberg-Essen, said, adding that she was protected by an “armour of popularity”.
But he added: “The situation is not flattering for the chancellor, a loss of power is quite evident”.
Peter Altmaier, Mrs Merkel’s national refugee coordinator and the head of her chancellery office, tried to downplay the disputes with Mr de Maiziere as a “communication misunderstanding”.
“It is clear there are a lot of discussions over this issue,” he told the talk show. “I hope that we can discuss this internally and behind closed doors. However it is vital that we act as one – as we do.”
Mr Schäuble has come under fire for his intervention from coalition partners, and from Joachim Gauck, the country’s usually non-political president.
Mr Gauck broke with protocol to warn against those who “voice assumptions and perpetuate stereotypes”, in remarks widely seen as directed at the finance minister.
Mrs Merkel came under intense questioning in a special half-hour interview on ZDF television entitiled What Now, Mrs Merkel? on Friday evening.
In the interview, she vowed to continue her “open-door” refugee policy: “It is our principle to help people in need,” the German chancellor said. “We need to show the freedoms we enjoy in practice and help those in need.”
“I cannot unilaterally define a limits. We in Germany cannot simply determine unilaterally who can come and who cannot.”
Angela Merkel
Mrs Merkel dismissed claims that her government was in crisis at the end of a week that has seen two of her most senior ministers openly challenge her refugee policy.
“The chancellor has the situation under the control, the federal government has the situation under control,” she said.
“I am sure that we will continue to show a friendly face. That is my sort of welcoming culture.”
But she refused to give way on her insistence that Germany can handle a record influx of some 800,000 asylum-seekers this year.
Asked about her earlier slogan of “We can do it”, she replied: “I think we have to work to make sure we can do it, and I believe we can do it.”
She refused to set a limit for the number of refugees Germany could take in.
“I cannot unilaterally define a limits,” she said. “We in Germany cannot simply determine unilaterally who can come and who cannot.”
Mrs Merkel refused the discuss the alleged rebellion. “Wolfgang Schäuble is in a class of his own,” she said enigmatically.
The chancellor admitted she had made mistakes in the past and said it was up to her to reduce the numbers of asylum-seekers and to crack down on illegal immigration.
But she vowed Germany would continue to open its doors to genuine refugees.
“I’m not the first chancellor to fight for something,” she said, comparing her situation to that of Helmut Kohl, the chancellor who oversaw the reunification of Germany.
She was critical of Germany’s European partners, saying she regretted the EU had not been able to come up with a common solution.
She compared the numbers arriving in Europe to the more than two million Syrian refugees Turkey has taken in, including some 900,000 children.
“We are a much richer continent than Turkey, I think it’s obvious,” she said. “We talk about human dignity, then we say we have be careful about refugees.”
While ministers argue, civil servants at the federal office for migration and refugees have published an open letter warning of serious flaws in procedures.
Asylum-seekers are being accepted as Syrians without being asked for any proof of their nationality, they warned.
Those claiming to be Syrian do not need to show passports, according to the letter. The only checking of their identity is carried out by freelance translators who often have little or no experience of Syrian dialects or accents, and are not accountable for any mistakes.
A “large proportion” of asylum-seekers were giving false identities in order to stay in Germany, the letter warned.
“The discontinuation of identity checks has also facilitated infiltration by Islamic State of Iraq and the Levant terrorists into Europe,” it claimed.
Austria on Friday announced plans to build a 2.5-mile stretch of fence on either side of its busiest border crossing with Slovenia.
Hungary and Slovenia have already built fences along sections of their borders.
Austrian officials said the new fence was not intended to prevent asylum-seekers from entering the country, but to control the flow.
“It’s about an orderly entry, not a barrier,” Josef Ostermayer, a minister at the Austrian chancellery said.
Meanwhile in eastern Germany, an eight-month pregnant asylum-seeker from Somalia was attacked and badly beaten.
The 21-year-old woman was taken to hospital. She has not been named and details of her injuries have not been released.
Police said they suspect two boys aged 14 and 15 and a 14-year-old girl of carrying out the attack.