Now, two top EU foreign-policy officials, EU External Relations Commissioner Benita Ferrero-Waldner and foreign-policy chief Javier Solana, have become directly involved in what increasingly looks like a mediation effort.
Solana is in Tbilisi on June 5 and from there heads to Sukhumi, the de facto capital of Abkhazia.
The region also figured prominently during June 4 talks in Moscow between Ferrero-Waldner and Russian Foreign Minister Sergei Lavrov.
Speaking at a conference at the European Parliament on June 5, Ferrero-Waldner said the issue was "very, very high" on her agenda during talks with Russian officials.
Her main objective, she said, was to push for a fresh dialogue -- "on the one hand between Georgia and Abkhazia, and on the other hand between Georgia and Russia."
Ferrero-Waldner was clearly trying to highlight the positives, noting that Georgian President Mikheil Saakashvili and his Russian counterpart Dmitry Medvedev have agreed to meet for face-to-face talks on the sidelines of the June 6-7 informal CIS summit in St. Petersburg.
She said this could "hopefully" lead to a new Georgian-Russian dialogue. She did warn that such dialogue would be "very complicated," adding that both parties must be prepared to compromise.
Ferrero-Waldner also hinted that the EU has begun exploring a possible compromise for a critical stumbling block in the Abkhaz dispute -- the continued deployment of Russian peacekeepers in the conflict zone.
The peacekeepers have been based in Abkhazia as part of a cease-fire agreement that ended a civil war in the early 1990s. But Tbilisi has long objected to the presence of Russian troops, particularly in recent months as Moscow has openly sought to strengthen ties with Sukhumi.
Moscow has stoked tensions further in recent weeks by sending additional troops to Abkhazia -- most recently deploying 400 soldiers with the specified intent of repairing Abkhaz railway links.
Tbilisi says such moves violate the terms of the cease-fire agreement and represent a direct violation of Georgia's sovereignty and territorial integrity.
The European Parliament effectively backed Georgian concerns, saying in a resolution passed on June 5 that Russia's peacekeeping force in Abkhazia is no longer "neutral or impartial" and calling for its UN mandate to be renegotiated.
The resolution is nonbinding, however, and will have no direct impact on EU foreign-policy decisions, which rest in the hands of the bloc's member states.
Threat Of Violence?
Ferrero-Waldner said she and Russian officials discussed the possibility of "changing here and there the [composition] of the peacekeeping troops, maybe to have some other CIS troops."
She described her Moscow counterparts as "totally against" the idea, adding that they wanted to see a "clear statement or declaration" committing Georgia to nonviolence.
A new report by the International Crisis Group, however, suggests that Georgia may itself be considering a military option in resolving the Abkhaz crisis.
The June 5 report claims Georgia has "quietly been making military preparations, particularly in western Georgia and Upper Kodori," a region that serves as the de facto border between Abkhazia and the rest of Georgia.
"A number of powerful advisers and structures around President Mikheil Saakashvili appear increasingly convinced a military option in Abkhazia is feasible and necessary," the report says in its executive summary.
"The Georgians have been warned by their Western partners against attempting a military solution," it adds. "But there are strong feelings in Tbilisi that something must be done to change a status quo in which Russia challenges the country's sovereignty with virtual impunity."
Giga Bokeria, Georgia's first deputy foreign minister, dismissed the claim. "Georgia is not contemplating solving this problem by force -- in fact, Georgia is doing everything possible to prevent such a scenario from materializing," he told RFE/RL's Georgian Service.
"However, this can't happen at the expense of conceding our territory or sovereignty. The Georgian people, Georgian society elected this government to defend our country's interests," he continued. "Not a single democratic government could accept what has been happening [in Georgia] in recent days."