The 23-year-old centre-back started his career with AZ Alkmaar and joined the Biancocelesti on a free transfer in the summer of 2015.
It was reported by Sky Sports News last week that the Saints were in advanced discussions over Hoedt's signing, and Tare announced the move on Sunday.
Tare told Italian broadcaster Premium Sport: "He requested to be transferred. He wants to be a starter in England and will be transferred to Southampton in the near future.
"He has had a positive experience here in Rome. I wish him all the best."
Media reports suggest that Southampton offered Lazio an initial £15.5m for Hoedt - a Netherlands international like wantaway defender Virgil van Dijk - but that the final deal is worth around £17m.