Foreign Minister Sebastian Kurz said that he was "very confident" that the request in the form of a letter would have the desired effect.
Albania was originally supposed to be given the status of an accession candidate last December, but was blocked by Britain, Germany and Denmark.
At an international Balkan conference in Vienna on Tuesday, Austria started an initiative to get Albania closer to its accession.
Aside from Kurz, the letter was signed by other EU foreign ministers attending the conference - Vesna Pusic for Croatia, Karl Erjavec for Slovenia, Miroslav Lajcak for Slovakia, Kristian Vigenin for Bulgaria, Evangelos Venizelos for Greece and Federica Mogherini for Italy.
The missive will be presented to the other EU foreign ministers at the next meeting in Luxembourg on June 23rd and 24th.
The EU is keen to encourage further reform in Albania, particularly as regards stamping out organised crime and corruption.