The YouGov survey in Britain, France, Germany, Denmark, Sweden and Finland showed a majority of those questioned backed action by Western powers against President Bashar Assad’s government.
But about half of Danish, Swedish, British and Finnish respondents wanted UN approval first, with more than a third in Germany and France thinking likewise.
Russia and China have blocked two resolutions condemning Assad over the bloodshed in Syria at the UN Security Council and so far Western powers have been wary of advocating military action.
Assad’s army bombarded opposition targets in the flashpoint city of Homs on Monday despite a promise to UN peace envoy Kofi Annan a week ago to cease fire and withdraw.
Assad has accepted but not implemented Annan’s six-point peace plan, which would allow Syria’s president to stay in power.
The poll showed support for intervention without UN authorization was strongest in Sweden (30 percent), France (29 percent) and Germany (27 percent).
Respondents in Britain, which like France sits on the Security Council, were the least happy with this option, with only 10 percent in favor of action without UN approval.
The UK also showed the highest proportion - at 24 percent - against military intervention under any circumstances. In Finland the equivalent figure was 18 percent, in Germany 16 percent and in Denmark 14 percent.
The most popular course of action was for Western countries to only intervene with UN backing. In Denmark this drew 53 percent of respondents, 48 percent in Britain, 45 percent in Finland and Sweden, 39 percent in Germany and 35 percent in France.
The YouGov study was carried out from March 22-27 and questioned 3,403 French adults, 1,734 Britons, 1,010 Germans, 1,007 Swedes, and 1,004 Finns.
