Jewish sources clearly indicate the Ark as being hidden under the Temple mount.
Avos D’Rebbi Nosson 41:12
These things were made and are hidden away: The Tent of Meeting and its utensils, the Ark and the broken Tablets, the flask of Manna, [a flask of anointing oil], the staff of Aharon with its blossoms and flowers, the clothes of the High Priest and Anointed Priest. However, the mortar of the House of Avtinas, the Table, the Menorah, the Curtain and the Tzitz are still in Rome.
Maimonides, Mishneh Torah, Laws of the Temple, 4:1
There was a stone in the Holy of Holies in the West that the Ark was placed upon. In front of it was the flask of manna and the rod of Aharon. When Solomon built the Temple and knew that it would be destroyed eventually he built a place hidden and deep to hide the Ark underneath the Temple. King Josiah commanded that it be hidden in the place that King Solomon had built…
The staff of Aharon, the flask of Manna, and the anointing oil were hidden with it and none of these things were returned during the Second Temple era…
However there are sources that indicate that some of the treasures were indeed taken to Rome.
Midrash Tanchumah (Buber) Vayakhel 10
Tosefta Yoma 2:14
Talmud Yoma 57a
Rav Elazar ben R. Yossi said, “I saw [the Parochet] in Rome and there were on it a number of drops of blood.”
Talmud Sukkah 5a
Rabbi Elazar ben R. Yossi said, “I saw the Tzitz of the High Priest in Rome and ‘Holy unto God’ was written on one line.
Avodah Zarah 11b, Rashi ad loc.
The [embalmed] face of Rebbi Yishmael the High Priest –Rashi – and it is still found in the treasure houses of Rome
Midrash Rabba, Esther 1:12
I saw the broken pieces [of the royal throne] in Rome
Most likely is that the treasures in Rome were either looted during the sacking of Rome, some were probably melted down and sold to finance the building of the Colloseum, and some may still be in the Vatican, which is the "heir" of the Holy Roman Empire, and ultimately of Rome. As to getting the stuff back, as they say in NY "fuggedaboudit". The Vatican will not even admit to having any of it and will not allow free access to its vaults. I have no idea how, or whom to contact at the Vatican, I suggest finding the Vatican website and making enquiries directly with them.
