The measure, proposed by the bloc's diplomatic service, would force Russian diplomats stationed in an EU member state to notify another member state if they wanted to travel there. That would give governments the option of barring them from entering.
Diplomats said the proposal had garnered broad support but it would only be clear if an agreement could be reached when member states' ambassadors to the EU meet later this week.
"Things are going quite well. But it's not a done deal yet," one EU diplomat said on condition of anonymity regarding the confidential discussions.
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The EU's External Action Service last month added the proposed restriction to a 19th package of sanctions against Russia over its war in Ukraine.
Countries are still negotiating the overall package, which includes bringing forward an import ban on Russian liquefied natural gas imports by a year.
The negotiations have been complicated by demands from Austria to unfreeze some Russian assets as compensation for penalties that its bank Raiffeisen faces in Russia.