Throughout the 2023-24 Premier League season, the 25-year-old full-back struggled to secure a regular starting place under Ange Postecoglou.
The Brazilian international made just 22 league appearances for the Lilywhites last term, starting only 11 times and mainly serving as a backup in defence.
Pedro Porro was preferred as Postecoglou's first-choice right-back last season and there is widespread expectation that he will remain the top pick next term as well.
Royal is currently in Japan with the rest of the Tottenham first-team squad, preparing for Saturday's pre-season friendly against Vissel Kobe in Tokyo.
Milan preparing improved bid for Emerson Royal
Milan are actively looking to bring in a right-back this summer, and have shown interest in the Brazilian, but negotiations reportedly stalled earlier this month when Spurs rejected their opening offer.
Premier League Brasil suggested that Paulo Fonseca's side initially tested the water with a bid in the region of £8.5m for the Spurs right-back earlier this month.
However, according to transfer expert Fabrizio Romano, a new round of talks between Tottenham and the Rossoneri have now commenced with the sides closing in on an agreement.
The report adds that there is now just a £2.5m difference in the valuation of the player by the two clubs and discussions are ongoing in the hope that a deal can be reached in the coming days.
It is understood that the Serie A giants have already made contact with the defender and agreed personal terms regarding his potential move to Italy.
Royal's current contract with the North Londoners runs until the summer of 2026 and Milan are believed to be prepared to offer the 25-year-old a five-year deal if he makes to move to the San Siro.
Busy summer at Tottenham Hotspur Stadium
Spurs are considering selling several players this summer with midfielder Pierre-Emile Hojbjerg recently transferred to Marseille and Royal might be the next to depart from the North London club.
Postecoglou's side have already confirmed Timo Werner on another loan and spent nearly £40m on Archie Gray and Lucas Bergvall. Bergvall officially joined the club this summer after finalising his move back in January.
The Brazilian is believed to be eager to move to the San Siro, and reports suggest he may have rejected other offers, including those from Saudi Arabia.
If the deal with Milan for the set price gets over the line this summer, Spurs will be able to recoup almost the entire £21.1m they spent on the 25-year-old when they signed him from Barcelona in 2021.
Royal has made a total of 101 appearances for Lilywhites over the past three seasons, notching up four goals and two assists during his time in the Premier League.