Midfielder Vuckic, 21, has represented Slovenia at senior international level and has also featured for the Magpies first team on 15 occasions.
Rotherham manager Steve Evans said: "This is a fantastic signing and he is a player we look forward to working with. He certainly adds a lot of quality to a very talented squad.
"I am grateful to Alan Pardew and Newcastle United Football Club. The lad will train with us and, all being well, will be in contention to make his debut on Saturday."
Right-back Tavernier, who has made two first-team appearances for Newcastle, is also expected to make his Millers debut in Saturday's clash with Gillingham.
Both players will remain at the New York Stadium until the New Year.