The 30-year-old England international is currently on a half-season loan at Newcastle United.
But in recent weeks, Lazio have been in touch with Rose’s camp to understand the player’s intentions.
The Biancoceleste are ready to invest in their left side with 34-year-old Senad Lulic becoming more injury prone.
Lazio’s other left-back Jordan Lukaku is Lulic’s heir apparent but has also struggled with injuries.
Rose has appeared in the last four editions of Europe’s top competition with Spurs and fits the bill.
With his contract expiring in 2021, Rose could be available for around €5.5m (£4.96m).
Lazio have yet to make a firm decision on whether to move for Rose, but are following him with interest.