The dpa news agency, citing information from unidentified participants, reported that a almost eight-hour meeting of the CSU in Munich was put on hold after interior minister Horst Seehofer made his offer.
She also pledged to conclude bilateral agreements with European partners such as Greece and Spain, with the goal of returning asylum seekers, who first entered the EU soil from these countries, but later arrived in Germany.
Mrs Merkel has insisted on Europe-wide solutions to handling the waves of foreigners trying to reach the continent and the standoff could spell the end of her fourth government.
Merkel fears Germany unilaterally setting up border checks could cause a domino effect across the European Union, undermining the Schengen free-movement zone and damaging the economy and political solidarity.
Horst Seehofer, leader of the CSU, a key party in Merkel's coalition, has become increasingly hostile to her position on immigration.
The currency then bounced on news Seehofer had offered to step down as minister and as chair of his Christian Social Union (CSU) party.
In a meeting with his party leaders on Sunday night, Seehofer offered to resign from his ministerial role and party leadership, reports said.
Both the CDU and the CSU are holding separate meetings Sunday to discuss Merkel's latest efforts on migration and plot their next steps.
In a document sent to leaders of her coalition partners, seen by Reuters, Merkel listed 14 countries, including some of those most outspoken in their opposition to her open-door refugee policy, which had agreed to take back migrants.
A Forsa poll showed that 67 percent of Germans thought the behavior of Seehofer and the CSU was irresponsible.
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Germany's interior minister has offered to resign over Angela Merkel's European Union deal to tackle immigration, reports say.
Senior CSU lawmaker and former interior minister Hans-Peter Friedrich wrote on his Twitter account that the CDU and CSU wouldn't let themselves be divided.
In an extremely fluid situation, Merkel and Seehofer are due to meet later in the day and before that, a joint session of the parliamentary groups of the CSU and CDU is due to take place, although there was some doubt if it would go ahead.
She could call a confidence vote in the Bundestag, which could trigger fresh elections.
The source of the impasse is a migration plan hatched by Seehofer to take unilateral action to turn back migrants who have been registered first in another European Union country. "We will stay together", he said to several minutes of applause, according to participants.
The two center-right parties have been in an alliance for 70 years under which the CDU left the CSU to fly the conservative flag in Bavaria.
The CSU party, facing a stiff challenge in Bavaria from the far-right Alternative for Germany (AfD) party in October's regional election, is determined to seal off its right flank.
As the CDU and CSU parties hunkered down in Berlin and Munich, a compromise appeared ever more elusive.
"As a result Italy will be in the position to have really good reasons for no longer accepting any people coming to Italy by the Mediterranean", Patzelt says, adding that this would be exactly the result the CSU intends.
Its success was based on strident anti-immigration rhetoric, attacking Mrs Merkel's decision to let in more than a million asylum seekers in 2015-2016.