German Chancellor Angela Merkel is willing to welcome "hundreds of children and their families, perhaps even thousands," Germany's top-selling Bild daily reports.
The German government has said it was in talks to take in families with children left homeless after a huge fire devastated the Moria migrant camp on the Greek island of Lesvos.
Germany has already signed up to a European scheme to host unaccompanied minors from the camp, promising to take in around 150 of the 400 arrivals.
Speaking to reporters in Berlin, Chancellor Angela Merkel's spokesperson Steffen Seibert called the help for the minors "a first step" but said more needed to be done to alleviate the suffering of some 12,000 migrants sleeping rough since last week's blaze.
"Talks are now ongoing in the federal government about how else Germany can help, what other substantial contribution our country can make," he said.
A "second step" would focus on families with children from the camp, Seibert said, echoing earlier comments by Interior Minister Horst Seehofer.
The spokesperson declined to put a number on how many families Germany might take in.
Germany's top-selling Bild daily, citing government sources, said Merkel was willing to welcome "hundreds of children and their families, perhaps even thousands."
German media reported that a deal could be agreed as early as Wednesday, when Merkel holds her regular cabinet meeting.
Finance Minister and Vice-Chancellor Olaf Scholz separately said that Germany, as the bloc's biggest country and economic powerhouse, "must take a substantial second step" when it comes to helping the Moria migrants and refugees.
"I want to clearly convey ... that we will come to an agreement."
Germans are still deeply polarised by Merkel's 2015 decision to keep the borders open to allow in hundreds of thousands of asylum seekers, and plans to take in more people from Moria are likely to stir debate.
At the same time, a string of German cities and regions have said they are prepared to house migrants from Moria, piling pressure on the federal government to do more.
Seibert on Monday called the destruction of the Moria camp, in an apparent arson attack, a "humanitarian emergency, a one-off emergency situation."
Lesvos is the main port of entry for arrivals in EU member state Greece because of its close proximity to Turkey.
The Moria camp was several times over its official capacity, leaving residents living in dire conditions.