It was announced last week that Robertson will leave Liverpool this summer as his current contract runs down.
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The Scotland captain was blocked from joining Tottenham in January and is now in talks about a summer move.
BBC Sport says an agreement is close to being struck on contract terms between Robertson and Spurs management.
However, any move will hinge on Spurs avoiding relegation.
Tottenham sit 18th on the Premier League table and are in the relegation zone, two points adrift of 17th-placed West Ham.