The issue has vexed the 28-nation bloc for years but gained prominence after the sudden influx of hundreds of thousands of migrants from outside Europe in 2015.
Germany took in the bulk of those migrants but has insisted that other countries need to contribute too, either by accepting migrants or contributing funds.
Failure to find a solution has given ammunition to Merkel's domestic critics.
Merkel, who is running for a fourth term in September's general election, said today she would insist on "a bit of solidarity" from all EU states but "how this will look exactly is something I can't say at this point.