The ministry's proposal comes in the run-up to the 10th anniversary of the 2008 war between Russia and Georgia.
International attention has lately focused on the killings of two Georgian citizens, Giga Otkhozoria and Archil Tatunashvili, in Georgia's occupied regions.
The Foreign Ministry said in a press release that Georgian officials continue to be impeded in their efforts to investigate the crimes, adding that human rights violations by Russia "undermine security and create tensions in the region".
The Georgian government in June announced sanctions against individuals under the so-called Otkhozoria-Tatunashvili list.
"In view of the request from the government of Sakartvelo, the Ministry of Foreign Affairs of Lithuania on August 2 turned to the Migration Department, proposing that it should also apply Lithuanian national restrictive measures against these individuals," the ministry said.
Sakartvelo is an alternative name for Georgia used in Lithuania.
Russia keeps thousands of its troops at military bases in the regions on a permanent basis. The international community considers Abkhazia and South Ossetia as occupied territories of Georgia.
Georgia's Otkhozoria-Tatunashvili list includes 33 names, including Alik Taboyev, head of the South Ossetian KGB in Akhalgori, and Akhalogori Deputy Prosecutor David Gurtsiyev.