Gian Piero Gasperini's side moved quickly to secure Oosterwolde, a versatile 25-year-old Dutch defender capable of playing at either full-back or centre-back, with the Turkish club agreeing to a loan carrying a €16 million option to buy.
That fee becomes an obligation if certain conditions are met, tied to Oosterwolde's individual performances, AS Roma's league standing and the club's progress to the Champions League knockout stages.
The player, who previously spent time on loan at Parma, has agreed a four-year contract with a fifth-year option worth €1.6 million net annually, and is expected to be sidelined until mid-September with a muscle issue picked up before the move.
The Oosterwolde deal came together as Roma's pursuit of Bynoe-Gittens hit a firm wall in London.
According to Sky Sport, Chelsea have rejected an approach structured as a loan with an obligation to buy, with the west London club unwilling to part with the 22-year-old at this stage of the window.
Manager Xabi Alonso is understood to have been central to blocking the move, viewing the England youth international as an important part of his squad going forward and effectively vetoing his sale.
Gasperini had identified Bynoe-Gittens as a priority to add further attacking width, and Roma will now have to decide whether there is room to reopen talks with Chelsea before the deadline or pivot to alternative targets on the wing.
Oosterwolde's arrival, meanwhile, could pave the way for a separate departure in defence, with Roma seeking around €20 million for young Poland international Jan Ziółkowski amid interest from Saudi club Neom, as the club continues to reshape its squad in the closing days of the market.