The sum will be a "real budget with annual revenues and spending", said French President Emmanuel Macron after talks with the German leader, adding that Paris and Berlin hoped to have it in place by 2021.
A defiant Interior Minister Horst Seehofer warned that he would give Merkel a fortnight to find a European deal to curb new arrivals by a June 28-29 EU summit, failing which he vowed to order border police to turn back migrants.
News agency dpa reported that Bavarian governor Markus Soeder said Wednesday: "We can not launch extra shadow budgets now or try to weaken the stability of the currency".
Referring to hasty arrangements and a domestic crisis over migration policies within Merkel's coalition government, the fervently anti-migrant Hungarian prime minister, Viktor Orban, said: "We understand that countries have domestic political difficulties, but this can't result in pan-European confusion".
Far-right Interior Minister Matteo Salvini also reacted furiously in remarks reported by Italian media.
European countries would take in migrants who qualify to apply in a shared plan that lifts the burden of carrying for them from the Mediterranean nations on the front line such as Italy, Spain and Greece, Macron said.
At the heart of the problem lie deep divisions over who should take responsibility for arriving migrants - often Mediterranean countries like Italy, Greece and increasingly Spain - how long they should be required to accommodate them, and what should be done to help those European Union countries hardest hit.
"We can not take in one more person", Salvini told the German weekly Der Spiegel.
Pope Francis told Reuters in an interview that populists were "creating psychosis" on the issue of immigration, even as ageing societies like Europe faced "a great demographic winter" and needed more immigrants.
Merkel, whose political capital is depleted after her party lost support in a September election, is in her fourth term in office and heads an uneasy alliance between her conservative CDU, its Bavarian counterpart, the CSU, and the centre-left Social Democrats (SPD).
Merkel spokesman Steffen Seibert dismissed the suggestion that the chancellor was trying to buy deals on migrants with financial concessions.
Migrants not entitled to asylum should be returned directly to their country of origin and not via other countries, Macron said.
The two leaders had chose to focus on broad issues to leave more room for negotiations with the other 17 euro members, Macron said - a scenario that leaves scope for the plans to be watered down.
Migrants seeking to reach Europe were picked up in their hundreds in the Mediterranean on Friday and Saturday.
Malta - which is also refusing to take in the boat in a new diplomatic standoff with Italy - nevertheless said it had sent in humanitarian supplies. "Hopefully a joint position that goes in the right direction".
German Chancellor Angela Merkel speaks during a news conference at the Royal Palace in Amman, Jordan June 21, 2018. "Simply because it's a form of additional budget", Bavaria's state premier Markus Soeder told Sueddeutsche Zeitung.
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