Saints have been in talks to sign the 23-year-old Netherlands international centre-back for the past few days.
Reports in Italy claim Southampton offered an initial £15.5 million but Sky Italia are now stating the final deal is worth £17 million to the Roman club.
Tare told Italian broadcaster Premium Sport: "He requested to be transferred. He wants to be a starter in England and will soon be transferred to Southampton.
"He has had a positive experience here in Rome. I wish him all the best."
Hoedt sat on the bench as an unused substitute as Lazio opened their Serie A campaign with an uninspiring 0-0 draw with newly-promoted SPAL on Sunday.