Dr Merkel said euro zone reform was the toughest issue in their talks at her Meseberg retreat outside Berlin, where they also touched on European foreign and defence policy and immigration.
"We want to support Italy in its call for solidarity and hope that Germany will also be met with understanding when it comes to solidarity in Europe with the issues of migration".
However, Merkel has been unyielding in defending her initial decision to keep Germany's borders open, telling lawmakers earlier this month that "in an exceptional humanitarian situation, Germany behaved very responsibly".
The agency, which cited unnamed participants at an ongoing meeting in Munich of the leadership of Seehofer's Christian Social Union party, said that if no agreements with other European Union partners are reached by then, the idea would be to start turning back some migrants at Germany's border.
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". "Merkel's argument for Europe is substantial - but not popular".
Berlin is also reportedly preparing to call a meeting between Merkel and the leaders of several European Union frontline nations in the migrant crisis ahead of the European Union summit.
Since 2016, numbers of asylum claims have fallen sharply, with a 44 per cent reduction in 2017 in applications from the previous year.
With just a week until what promises to be a contentious European Union summit, the leaders of the bloc's two powerhouses were keen to put on a show of common objective. "Big mistake made all over Europe in allowing millions of people in who have so strongly and violently changed their culture!" Merkel has common sense on her side when she argues that there has to be a European fix.
On Sunday, leaders in Trump's own party and Democrats called for the end of the president's practice of separating children from their parents when families arrive at us borders seeking entry. But the crisis shaking the new administration has come from within Merkel's own conservative family, as her interior minister has pushed for Germany to turn back unilaterally some migrants at its border.
Italy and France have also become involved in the migrant debate, with Italy's interior minister Matteo Salvini, who is the deputy Prime Minister of the country, refusing to accept new refugee boats and calling for asylum seekers to be equally distributed across the European Union, rather than being hosted by the first country they land in. If that were to happen, Mayer says, Merkel would have the authority to fire Seehofer.
Merkel said the primary objective of the budget, which could go into effect in 2021, would be to fund "investments and convergence measures".
The CSU, which faces regional elections in October, fears it could be toppled from its decades-old perch atop the wealthy southern state by the anti-immigration Alternative for Germany (AfD) party if it does not tighten policy towards migrants.