The 6ft 2in defender came through the youth ranks at Hellas Verona before joining Udinese on loan last year, and Le Zebrette made the deal permanent for £3.6m only a few weeks ago.
Udogie immediately cemented his place in the Udinese XI - often featuring as a left-wing back - and he amassed five goals and three assists from 35 Serie A appearances in the 2021-22 campaign.
The teenager has also won four caps for the Italy Under-21 side and remains under contract at Udinese for another four years, but his spell with Le Zebrette could come to a swift end.
According to Calciomercato, Tottenham and Udinese have a verbal agreement in place for the transfer of Udogie, who will cost the Lilywhites a total of €25m (£20.9m) including add-ons.
The report adds that Tottenham will allow Udogie to return to Udinese on loan for the 2022-23 season to further his development, and Spurs have been in contact over a deal for a month now.
Udinese were determined to keep hold of the highly-rated youngster, but Udogie has already given the green light to a move to Tottenham, who have fended off competition from numerous other clubs.
Inter Milan allegedly tried and failed to sign Udogie, while Juventus, Lazio, Manchester City and Brighton & Hove Albion were all keeping a close eye on his development in Italy.
Udogie would follow a long list of Serie A players to make the move to North London since the arrivals of Fabio Paratici and Antonio Conte, including summer signing Ivan Perisic.
Cristian Romero, Rodrigo Bentancur and Dejan Kulusevski have also become integral cogs in the Lilywhites XI, although Pierluigi Gollini failed to impress during his loan spell.
Tottenham have already wrapped up the signings of six players this summer, with Perisic joining Fraser Forster, Yves Bissouma, Richarlison, Clement Lenglet and Djed Spence in North London.
The Lilywhites are expected to focus on outgoings for the remainder of the window, having already allowed Steven Bergwijn, Cameron Carter-Vickers and Jack Clarke to leave.
The likes of Giovani Lo Celso, Tanguy Ndombele, Bryan Gil and Sergio Reguilon could all depart in the coming weeks, and Joe Rodon has signed for Rennes on loan with an option to buy.
Tottenham will commence their 2022-23 Premier League season at home to Southampton on Saturday, but Richarlison will not be available due to suspension.