Standard Sport understands that Spurs will pay £15million up front for the teenage left-back, a fee which could rise to £20m with add-ons.
Udogie is set to agree personal terms and undergo a medical in Italy, where he will remain for the 2022-23 Serie A campaign on loan back at Udinese. There are no plans to bring him to their Hotspur Way base in order to conclude a deal.
The highly-rated Italian Under-21 international, 19, will become Tottenham's seventh signing of a busy summer transfer window that has already seen Richarlison, Yves Bissouma, Ivan Perisic, Djed Spence, Clement Lenglet and Fraser Forster arrive in north London.
Udogie comes personally recommended by Tottenham boss Antonio Conte and is seen by the club as a young player with big potential, having made 35 appearances during an impressive campaign at Udinese, whom he joined on loan from Hellas Verona with an obligation to buy last summer.
The deal is similar to the one that saw Pape Matar Sarr join Tottenham last summer before being immediately loaned back to French club Metz for the 2021-22 season.
Spurs did not hand full debuts to any of their signings for the start of the latest Premier League season on Saturday, with Perisic and Bissouma coming off the bench in a 4-1 comeback thrashing of Southampton in N17.
Richarlison was banned for that fixture owing to a belated one-match suspension for a controversial flare celebration at Everton last term but will be available to make his Tottenham bow in the London derby against Chelsea at Stamford Bridge on Sunday.