The German Chancellor was granted a fortnight’s reprieve by critics within the Christian Social Union, led by interior minister Horst Seehofer, in order to try agree a solution with other EU nation states which would ease the burden on Germany.
Mr Seehofer’s party wants tougher immigration laws in the country, which has accepted more than one million migrants since 2015.
He is threatening to turn away migrants who are registered elsewhere in the EU at Germany’s borders.
Mrs Merkel met with other international leaders at a hastily convened “mini-summit” over the weekend which she hoped would break the deadlock prior to the full EU summit which starts on Thursday.
However, no solution was agreed by representatives of Austria, Greece, Malta, Bulgaria, Belgium, the Netherlands, Croatia, Slovenia, Denmark, Finland, Sweden, Luxembourg, Italy, Germany, France and Spain.
European Commissioner for Budget and Human Resources Mr Oettinger, who is a member of Mrs Merkel’s party, the Christian Democratic Union (CDU), warned: “I believe there will be progress by the time of the EU summit on Thursday and Friday, but there won’t be an agreement of the dimension that some in the CSU are expecting.”
Under the so-called Dublin rules, asylum-seekers need to be processed in the country in which they first arrive, usually Mediterranean countries like Italy, Greece and Spain.
In December EU leaders fixed a deadline of the end of June to reform the rules by establishing a permanent mechanism to relocate asylum-seekers throughout the bloc and setting time limits for how long a migrant must be in a country before they cannot be returned to where they first arrived.
However, Germans are increasingly opposed to Mrs Merkel’s open-door policy to asylum seekers, while new governments in Austria and Italy are also pushing back against proposed reforms to EU-wide migration policy.
Italy is vehemently opposed to reforms which could effectively mean it retains responsibility for migrants for ten years from the date when they first arrive in the country.
And Austrian Vice-Chancellor Heinz-Christian Strache (FPÖ) says Mrs Merkel has been weakened by the weekend’s debate.
He believes it could be “her political end”.
Mr Strache also demanded urgent action: “Either there is a European solution or we must ensure that people who have been rejected, don’t come back to Germany.”
He said if Germany stopped migrants there would be a “domino effect” with other EU states, beginning with Austria.
Mr Strache said: “Then there will finally be pressure so that the EU takes note that we have to initialise efficient external border control.”
If no deal is struck between EU member states at this week’s summit, Mrs Merkel may have to rely on bilateral talks with hardline nations such as Austria and Italy, with no guarantees of success ever then.
Leopold Traugott, a policy analyst with the Open Europe think tank, told Express.co.uk last week the collapse of Mrs Merkel’s government would have “terrible consequences for Europe”.
He added: “Should Merkel fail to reach any European deal – either at the EU summit, or in bilateral talks – we are back to the meltdown talk once again.
“The Bavarian CSU has announced it would in this case go ahead with its policy to reject certain asylum seekers at the national borders, which would put Merkel in a very delicate situation.
“Either she could reign Mr Seehofer in, thereby risking the unity of the government, or she could accept his unilateral action – which would be a sign of sheer powerlessness on her side.”
Meanwhile, Federica Mogherini, the high representative of the European Union for Foreign Affairs and Security Policy, is urging member states to support an African trust fund as a way of screening asylum seekers on the continent.
EU leaders are believed to be ready to back centres in countries including Algeria, Libya, Morocco and Tunisia.
(Additional reporting by Monika Pallenberg)