Britain has sought to persuade its EU peers since May to put the Shiite Muslim group's military wing on the bloc's terrorism list, citing evidence that it was behind a deadly bus bombing in Bulgaria last year.
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Until now, the EU has resisted pressure from Washington and Israel to blacklist Hezbollah, arguing that it could fuel instability in Lebanon, where the group is part of the government, and add to tensions in the Middle East.
Diplomats say the opposition to such a move is fading.
"There are still reservations, but we are moving towards what could be a decision on the possible listing," a senior EU official said.
"The number of member states which have difficulties with a possible decision has been slowly diminishing."
EU foreign ministers will discuss the issue on Monday in Brussels.
Blacklisting the military wing would mean the freezing of any assets it may hold in the 28-nation bloc, though officials say there is scant information on the extent of Hezbollah's presence in Europe or on its assets.
Britain, backed by France and the Netherlands among others, has argued that Hezbollah's growing involvement in the Syrian war means Lebanon is already in a fragile situation and that the EU must weigh the possibility of future attacks in Europe.
To soothe worries that sanctions against Hezbollah could complicate the EU's relations with the Lebanese government, EU governments are also likely to issue a statement pledging to continue dialogue with all political groupings in the country.
"A few member states wanted to be reassured that such a decision will not in any way jeopardize political dialogue," the senior EU official said.
Some EU diplomats, responding to concerns that sanctions could further radicalize the group, have argued that targeting the military wing could, in the long term, persuade some of its members to move away from violence into the political sphere.
Hezbollah denies any involvement in last July's attack in the Bulgarian coastal resort of Bourgas that killed five Israelis and their driver.
But the Bulgarian interior minister said last week that Sofia had no doubt the group was behind it.
In support of its bid to impose sanctions, Britain has also cited a four-year jail sentence handed down by a Cypriot court in March to a Hezbollah member accused of plotting to attack Israeli interests on the island.
Hezbollah was set up in Damascus by Iran in 1982 with the aim of fighting Israel after its invasion of Lebanon.
Its involvement in the Syrian conflict is widely seen as a major factor helping President Bashar Assad to withstand a two-year popular uprising led by the Sunni Muslim majority against his rule.
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