Envoys at Organization for Security and Co-operation in Europe headquarters in Vienna said that Martin Sajdik, currently Austria's ambassador to the United Nations, was in the frame for the job.
Austrian newspapers Die Presse and Kurier reported that Russian-speaker Sajdik, 66, was the favourite.
Kurier said that Austria's Foreign Minister Sebastian Kurz has already discussed it with his Russian and German counterparts Sergei Lavrov and Frank-Walter Steinmeier.
"He has good chances, but it's not 100-percent decided yet," one diplomat told AFP. Two others concurred but one said that Russia -- a member of the OSCE like Ukraine -- may object.
The Swiss government, which in January passed to Serbia the rotating OSCE presidency, said earlier this month that Tagliavini was stepping down but did not give a reason.
Her mandate had been renewed in January even though the presidency had passed to Serbia.
The 57-nation OSCE has a large observer mission in eastern Ukraine, where the UN says nearly 6,500 people have died in 14 months of fighting between Russian-backed separatists and Ukraine's military.
But the OSCE also chairs the contact group comprising Kiev, Moscow and the OSCE. Its next meeting with representatives of the rebels also present is set for June 16th in Minsk.