Merkel, however, opposes any unilateral move to reverse her 2015 open-door policy and undermine her authority.
A government spokesman said Merkel was also seeking talks with some European Union members before the leaders summit but denied a Bild report that she was trying to set up a special summit.
She suggested developing new formats, such as an "EU Security Council", and argued that in future decisions on foreign and security policy should be taken by majority vote, in order to increase the speed and effectiveness of the EU's decision making.
Macron echoed her words, saying that "a European answer" is needed to the challenges that migration poses.
Merkel's own position as chancellor appeared to be threatened after her hardline interior minister and conservative ally, Horst Seehofer, called for Germany to start refusing some refugees at the German border.
Seehofer struck a more conciliatory tone when he told Bild on Sunday that "it is not in the CSU's interest to topple the chancellor, to dissolve the CDU-CSU union or to break up the coalition".
The chancellor has flatly refused to back Mr Seehofer's plan to turn away migrants at the German border if they have already registered elsewhere in the EU.
Some 49% of voters surveyed last week by pollsters Civey for Die Welt newspaper said they either "definitely" or "sooner" place trust in Seehofer to solve the perceived asylum crisis compared with 32% who said Merkel has the right approach.
Speaking to dpa on Tuesday, Bavarian governor Markus Soeder (CSU) expressed skepticism that Merkel would succeed in meeting the tight deadline set by the CSU. On Monday, the CSU agreed to give Merkel two weeks to seek agreements with European partners.
Despite the migrant policy headache, Macron and Merkel might find more common ground on deeper European Union economic integration. And then: "We don't want what is happening with immigration in Europe to happen with us!".
Merkel stressed Monday that she doesn't want to see Germany unilaterally turn back migrants at its borders, as Interior Minister Horst Seehofer has been advocating.
In fact, crime in Germany has fallen to its lowest levels since 1992, even as the country has become home to a million refugees fleeing war-torn nations in the Middle East and North Africa.
The row with Bavaria's CSU, which faces a tough election in October, has piled the pressure on her to deliver.
In a tweet on Monday, Trump said that the people of Germany were turning against their leadership because of loose migration policies.
How it would be funded was also up for discussion with other eurozone members, said Merkel, suggesting that it could involve regular transfers made by individual countries or a tax on financial transactions. "Of course, we need to do more to combat crime, but those were indeed encouraging numbers".
Merkel has agreed on a common investment budget for the bloc, but says it should be worth several tens of billions of euros, not the hundreds of billions suggested by Macron.
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