"We will continue to work toward a solution".
Currently, frontline Mediterranean states, notably Italy and Greece, are grappling with the bulk of new arrivals, and asylum demands skewed toward western Mediterranean and richer northern European states.
Mr Sanchez said the meeting, involving the heads of 16 European Union countries, were "frank and open" but "we don't have any concrete consequences or conclusions". He said leaders planned to try to sort out the legality of a range of ideas.
It follows a heated exchange between Emmanuel Macron and Italy's new interior minister Matteo Salvini, who accused the French president of being arrogant and drinking too much champagne.
Macron also riled Italy, the main landing point for African migrants, by saying that the migration emergency, which peaked in 2015, had passed and was now mainly a political issue. Anti-migrant parties, and governments in Hungary and Italy, have been fomenting public fears of foreigners and have won support doing so.
Sanchez said it was "important we have a common answer to a common challenge". "When someone has the right to protection and asylum, we should grant it".
He said Europeans should "not forget our values".
Each of the member states, which include France, Italy and Portugal, will take "a few dozen individuals" of the 233 migrants on board the Lifeline vessel, Macron said in Rome after talks with Pope Francis.
Mrs Merkel, who is fighting a domestic political crisis with her coalition partners over migration, said she was seeking "bilateral and trilateral" deals to cope with short-term migration pressures.
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The International Organization for Migration (IOM) and the UNHCR are cautious about setting up "hotspots" outside of Europe to filter people fleeing violence at home from those trying to get to the continent to improve their lives economically.
Merkel stressed that European countries wouldn't be left alone in dealing with "primary migration" while others were left responsible for migration within the bloc. Unable to wait for full European Union consensus, Merkel said she would go for faster measures.
But Conte also wants to split responsibility of processing asylum claims of anyone disembarked from rescue boats in the Mediterranean Sea, in a nod to MS Aquarius, an NGO rescue ship operated by SOS Mediterranee and Doctors Without Borders.
But Hungary and Poland have refused to host any arrivals.
The leaders agreed that additional detention centres should be opened in Europe to review asylum seekers' applications.
Conte said such secondary movements would not be a problem if other European Union states help Italy more in dealing with arrivals.
After the summit, Mr Conte said it had gone well and "we are all pulling in the right direction".
Three years after a large influx of migrants and asylum-seekers, the vexed issue of migration has become a popular populist issue and threatens to topple the long reign of Europe's most prominent leader, Chancellor Angela Merkel of Germany. The decline is even starker compared with 2016: It equals just 19 percent of the same period in that year.
"I think it was better than expected, there was some progress that has been achieved", Maltese Prime Minister Joseph Muscat said, echoing a similar assessment by his Spanish counterpart, with both describing frank exchanges in the afternoon meeting.
The participating countries in Sunday's meeting were Germany, France, Italy, Spain, Greece, Malta, Bulgaria, Austria, Belgium, the Netherlands, Luxembourg, Croatia, Slovenia, Denmark, Finland and Sweden.