These sources said the exemption, which EU officials said they would monitor closely to avoid abuse, meant Sberbank and VTB subsidiaries could operate normally within EU member states.
The step will come as a relief to Austria, where the two big state-controlled Russian lenders have the headquarters of their European operations, the sources said.
Hungary and Slovakia had also said the latest sanctions - aimed at punishing Moscow for fomenting Ukraine's political crisis - should not apply to Russian banks operating in the EU to prevent disruptions on their retail operations, two sources close to the discussions told Reuters.
Details of the EU sanctions - which include banning state-controlled Russian banks from raising long-term financing by selling stock or bonds on EU markets - are due to be published on Thursday and enter into force on Friday.
The European Commission has not yet published a list of the Russian banks that will be affected but has said it will apply only to Russian banks that are more that 50 percent owned by the state or other public authorities. Sberbank is majority-owned by the Russian central bank, and VTB is majority-owned by the government.
"We limited the geographical scope to Russia. We did not want to include EU subsidiaries registered and established in the EU," one EU source said, seeing as the risk as "manageable" that these local lenders could transfer cash to their parents.
"These subsidiaries never issued corporate bonds or equities in the past three or four years, so it will be extremely visible if they start doing it now," the official said.
Sberbank Europe operates across central and eastern Europe after buying Austrian Volksbanken's VBI eastern European arm in 2012, and VTB Bank (Austria) has operations in Germany and France as well.
"We are not blind and deaf and stupid"
"When our bank supervisors looked into this they said: 'Listen, guys, what you do not want is to create a bank run,'" one government official said on condition he not be named.
He dismissed the suggestion that the move was an attempt to head off retaliation by Moscow against Austrian banks active in Russia such as Raiffeisen Bank International.
An EU sanctions committee will review how the system is functioning and ensure that EU-based subsidiaries are not funnelling funds they raise back to their parents. "We are not blind and deaf and stupid," the government official said.
Loans, including syndicated loans, are not covered by the sanctions but European banks will probably be reluctant to arrange lending to Russian borrowers.
Sberbank Europe and VTB Bank (Austria) declined comment. VTB, which will also be hit by U.S. sanctions that thwart its ability to access dollar financing, said earlier it was ready to borrow in other currencies and markets.