Asked in Berlin whether her government can work well until the end of its term in 2021 and whether she is still in full control, Merkel replied: "Yes to both".
Migration "is a European challenge that needs a European answer", she said in her weekly video message Saturday.
After the tweet, the president defended his stance during a press conference and repeated his claims.
The CSU leadership agreed on Monday to delay until after a June 28-29 European Union summit the introduction of an entry ban for refugees who have already registered in other EU countries, allowing Merkel time to reach a multilateral deal.
However, "the European project is at risk", and "we have a particular responsibility", she said.
There is no love lost between Seehofer and his party - facing a tough regional election in October - and Merkel, who have fought repeatedly on migrant policy in the last three years.
If no satisfactory European Union deal is reached, German police would start sending back migrants registered in other European Union states, it said, although CSU General-Secretary Markus Blume said the report was disinformation and: "There are no agreements in any direction".
German Chancellor Angela Merkel plans to discuss the issue of migration with Italy's new premier on his first official trip to Berlin.
"Whoever knows Europe, realizes this is no easy task", Merkel told reporters Monday, adding "we have a particular responsibility" to refugees.
Seibert declined to elaborate ahead of the meeting, which will be preceded by statements from the leaders.
After Merkel bowed to pressure from the CSU and made Seehofer her interior minister this spring, he told Bild: "Islam does not belong to Germany".
Seehofer is calling for Germany to turn away immigrants who have already registered in other European Union countries, which would undermine Merkel's open door policy adopted in 2015, since when the country has welcomed some 1.6 million refugees.
Seibert said such agreements could involve countries that are most strongly affected by migrant movements.
Over the past week, a conflict between Interior Minister Horst Seehofer and Merkel over migrant policy has escalated into a threat to her government.
Merkel and Macron agree that overburdened Mediterranean countries need financial support or incentives, and that the EU's Frontex border force must receive more funding, while Paris also advocates the creation of asylum processing centres in Africa.
Seehofer threatened Monday to turn back migrants if the German leader doesn't secure a deal within two weeks.
NELSON: The interior minister and his party, which is facing a major challenge from far-right populists in upcoming regional elections, say they will unilaterally prevent migrants from crossing Bavaria's worldwide borders if Berlin doesn't act.