The draft bill, which was endorsed by an overwhelming majority in Parliament yesterday (7 October), faces stiff opposition from France when it is submitted for approval by the 27 EU member states in the EU Council of Ministers.
"I now hope that we will have an agreement with ministers at first reading," said the European Parliament's rapporteur on the resolution, Slovenian MEP Tanja Fajon (Socialists & Democrats).
"This would allow citizens of the two countries to travel freely to spend Christmas with relatives elsewhere in Europe," she said.
France is opposed to lifting visa requirements for Bosnian and Albanian citizens. Paris is still suffering from the political repercussions of its controversial expulsion of illegal Roma migrants, some of whom were EU nationals of Romanian and Bulgarian origin.
French officials criticised the European Commission for pursuing visa liberalisation talks with Western Balkan countries for "political reasons" and neglecting the "risks" associated with opening up the EU's borders.
However, Paris does not have the power to block a decision by the EU Council of Ministers, which represents all 27 EU member states. Indeed, since the Amsterdam Treaty in 1999, justice and home affairs are considered a Community matter.
According to EurActiv sources, a few countries in the Council are supportive of France, but their votes will not suffice to block the decision.
The Netherlands and Denmark have frequently been mentioned as hostile to opening the Schengen visa-free area to Albania and Bosnia-Herzegovina.
Qualified majority voting (QMV) in the Council requires a minimum of 255 votes out of 345 (74.8%). The three countries opposed to the proposal have 49 votes in total and are far from creating a blocking minority.
"The French can do nothing. But they can suspend Schengen," MEP Charles Tannock (European Conservatives and Reformists; UK) told EurActiv.
Tannock was referring to the fact that Schengen countries have the right to temporarily introduce border controls by invoking a national emergency.
Tannock also believes France could delay the Council vote on political grounds.
"There is a problem, to be frank. I'm not absolutely happy. Albania has a lot of organised crime," Tannock said.