According to the latest speculation that comes courtesy of Turkish sources cited by the Daily Express, the Ligue 1 outfit are plotting an offer believed to be worth in the region of £14 million with the aim of bringing Portuguese U21 international Bruma to the Stade Louis II in time for the second half of the French domestic season.
The same source suggests that both aforementioned Premier League giants have been closely monitoring Bruma over recent months, with the 19-year-old having been commendably quick to demonstrate the full extent of his talent after moving to the Turk Telekom Arena from Sporting Lisbon last summer.
While he has has only made nine Super Lig appearances in total for Galatasaray, Bruma - who has also featured on six occasions in the Champions League for Cimbom - has obviously shown enough raw potential to convince visiting scouts that he has what it takes to thrive at a higher level.
Monaco's obvious ambition to strengthen their squad as soon as the winter window opens for business in the new year presumably comes at least partly as a result of criticism from fans following a difficult few months on the transfer front.
Having only lavished a considerable sum to bring James Rodriguez and Radamel Falcao to the principality from Porto a year ago, the club lost both world-class Colombian internationals last month as the former capitalised on his superb World Cup form to secure a dream switch to Real Madrid and the latter somewhat surprisingly joined Manchester United in an initial loan deal on deadline day.
Add those respective departures to that of new Newcastle striker Emmanuel Riviere and it becomes obvious that the current Monaco side is unquestionably weaker than last year's team that qualified for Champions League football after finishing eight points behind reigning champions Paris Saint-Germain.
Will Chelsea/United intervene?
At this stage and with several months remaining before clubs can buy and sell players once again, it is unclear if either Manchester United or Chelsea intend to intervene in Monaco's attempts to capture Bruma.
If £14 million really will be enough to convince Galatasaray to part with his services already, then it certainly could be worth at least pondering for both Louis van Gaal and Jose Mourinho although much is obviously likely to depend on how both clubs have progressed in the league up until that point.