One of Mrs Merkel's regional party leaders publicly questioned the government's economic policies.
There are calls for an emergency party conference to be held to address concerns.
Conducted by polling group Forsa, the opinion poll surveyed 578 members of Mrs Merkel's Christian Democratic Union (CDU) and its Bavarian sister party, the Christian Social Union (CSU).
It was carried out on 4 and 5 August, before the European Central Bank (ECB) agreed to intervene in the bond markets to support Italy and Spain.
Fewer than half of those surveyed supported the bailouts of Greece, Ireland and Portugal.
Christean Wagner, leader of the conservative bloc in the parliament of Germany's Hessen region, has demanded clarification of the Merkel government's policies at an emergency party conference.
Speaking to the tabloid Bild, he said there was a "groundswell" of demands from party members for a conference on the "policies and profile" of the federal government.
Similar comments were made in the same newspaper by the head of the CDU youth wing, Phillip Missfelder.
Mr Missfelder indicated that he would call for an emergency conference at the next meeting of the party's national executive, but only if Italy formally asked for help from the eurozone's bailout fund.
"The party has a right to participate in such a momentous decision," he said.
The current 440bn euros (£390bn; $632bn) made available by the European Financial Stability Framework (EFSF) is much less than would be required if Italy - or Spain - were forced to seek help from the facility.
At a summit in July, eurozone leaders agreed to give the EFSF new powers to make credit available to countries such as Spain and Italy that are not at immediate risk of insolvency.
The reforms will need parliamentary approval in the eurozone states next month.
Growing revolt from her own party may mean that Chancellor Merkel will be forced to rely on support from opposition parties to have the reforms passed.
Her conservative coalition with the liberal Free Democratic Party currently has a 21-seat majority in the German lower house of parliament, the Bundestag.