Sources have told ESPN that Tottenham will pay an initial £15m with a series of performance related add-ons worth in the region of £6m.
A medical is expected to take place in Italy this week.
Spurs have strength in depth on the left side of their defence already, with Ryan Sessegnon starting the season in impressive fashion as they thrashedSouthampton 4-1 in last Saturday's Premier League opener.
Sessegnon started as a left wing-back in head coach Antonio Conte's preferred 3-4-3 system, a role Ivan Perisic is expected to compete for following his arrival on a free transfer from Inter Milan.
Matt Doherty can also play there if required and so Udogie, 19, will spend the year continuing to develop at Udinese.