Antonio Conte is said to have personally recommended the Italy Under-21 international to Spurs, who would loan him back to Udinese next season.
Udogie is being tipped for a first call-up to the Italy squad following an impressive breakthrough season in Serie A.
The 19-year-old scored five goals in 35 league appearances for Udinese last season, having joined from Verona last summer.
Tottenham sporting director Fabio Paratici has good knowledge of the Italian market and Udogie is seen a young player with big potential.
The deal would be similar to the one Spurs secured last summer for Pape Matar Sarr, who joined from Metz before being immediately loaned back to the French club for the 2021/22 season.
Udogie would be the seventh new signing at Spurs this summer, following the arrivals of Ivan Perisic, Yves Bissouma, Richarlison, Clement Lenglet, Djed Spence and Fraser Forster.
Conte is happy with his options at left wing-back for next season after signing Perisic on a free transfer from Inter. Ryan Sessegnon will compete with the 33-year-old for a starting spot, with Sergio Reguilon told he can leave this summer.
Tottenham kick-off the new season at home to Southampton on Saturday and there is real excitement about their prospects as they target silverware under Conte.
Oliver Skipp is set to miss the start of the season after suffering a foot injury in training. The midfielder missed the end of last season with a pelvic issue and has been frustrated by his latest injury as he bids to establish himself in the team under Conte.
Richarlison is suspended for the Southampton game and fellow new boy Bissouma is a doubt after tweaking is hamstring in last weekend’s 1-0 friendly defeat to Roma in Israel.
