Tottenham have approached Manchester City about signing Savinho, with talks under way between the two Champions League clubs.
There are widespread reports that a fee of around £43m (€50m) has been discussed in these talks.
Sky Sports News understands the figure would need to be higher than that for City to consider selling the 21-year-old.
City are not interested in selling Savinho, who only signed for them last summer from sister club Troyes, and it would take a huge offer from Spurs for them to consider letting him go.
Any recruitment of wide forwards for Pep Guardiola in the final three weeks of the window is contingent on Savinho leaving.
Reports that the Brazil international's exit is imminent are believed to be premature.
Spurs have been in the market for a number 10, and that search is intensifying after a long-term injury to James Maddison.
Savinho can play at No 10 but is predominantly a winger.
Spurs winger Dejan Kulusevski has previously played at No 10 when Maddison has been injured.
The north London side have also lost forward Heung-Min Son this summer after his move to Los Angeles FC, bringing an end to the captain's decade at the club.
Savinho, who has mainly been used on the right wing at City under Guardiola, scored 11 goals for Girona in the 2023/24 season from the left side of their attack, and therefore could be a natural replacement for Son.
Savinho scored three goals and provided 13 assists in 48 appearances for Man City last season.
Savinho came on as a half-time substitute in Saturday's 3-0 pre-season win over Palermo.
Savinho may find game time harder to come by at City this season following the arrival of Rayan Cherki, while he faces competition from the likes of Phil Foden, Oscar Bobb and Jeremy Doku.
Tottenham, meanwhile, are pursuing a deal for Savinho, having spent £120.3m this summer on six signings.
The Europa League winners turned Kevin Danso's (£20.9m) and Mathys Tel's (£29.9m) loan moves into permanent deals.
Mohammed Kudus has been their most expensive signing of the summer so far after his £55m arrival from West Ham.
Midfielder Joao Palhiha has joined on loan from Bayern Munich, while defenders Luka Vuskovic (£9.5m) and Kota Takai (£5m) have also joined the club this summer.
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