Party sources told AFP news agency that Horst Seehofer, interior minister and leader of the CSU, told aides he was unhappy with the accord and complained to that he had endured a "conversation with no effect" with the chancellor on Saturday.
Pressure from the right is especially acute for the CSU as it faces October regional elections in Bavaria.
While most analysts expect Merkel to survive the clash with the CSU, it is unlikely to be the last occasion on which the sister party seeks to distance itself from a chancellor it sees as too centrist for its own supporters.
Mr. Seehofer had already offered his resignation late on Sunday evening during the CSU leadership's meeting.
In a televised interview on Sunday, Merkel said "The sum of everything we have decided has the same effect [as national measures]".
He said that "there is an abundance of possibilities ... for compromises", but didn't specify what they were. "In my view Europe will be held together, otherwise free movement could have been in danger", she added.
Merkel on Friday came away from an European Union summit with agreements from Greece and Spain to take back migrants previously registered in those countries, and an overall agreement by the 28-nation bloc to ease the pressures of migration into Europe.
At the heart of the dispute are Mr. Seehofer's plans - part of a 63-page "master plan" circulated to CSU leaders - to reserve the right to reject some asylum seekers, who have already registered in other European Union countries, when they reach the German border.
The chancellor had fought hard to reach a deal in Brussels.
"In the interest of the country and the ability by the coalition to act, we want to try to find a way to unify on this central question: border control and refusal, only on this question", Seehofer said early Monday.
Member countries could also create processing centres to determine whether the new arrivals are returned home as economic migrants or admitted as refugees in willing states.
Merkel also proposed that migrants arriving in Germany who first registered in another European Union country should be placed in special "admission centres" under restrictive conditions.
Secretary General of the Christian Democratic Union Annegret Kramp-Karrenbauer, center, arrives to deliver a statement after a party leadership meeting at the CDU headquarters in Berlin, on July 1, 2018.
According to the document seen by coalition sources, Merkel secured similar deals with a total of 14 countries including France and central European states that were fiercely critical of her migrant policies, such as the Czech Republic, Hungary and Poland.
Merkel's CDU relies on CSU to maintain power through a coalition, also including the Social Democrats (SPD), formed three months ago after an election in September.
Nevertheless, the anti-refugee, anti-Islam Alternative for Germany (AfD) surged into federal parliament for the first time past year, leading to months of paralysis while Merkel struggled to put together a workable coalition.