Nottingham Forest had already lodged a bid of €16 million for the 23-year-old English defender, according to Turkish outlet Sporx.
Fenerbahçe are holding out for a figure closer to €18 million and any formal decision on the player's future is expected to be made after the club's presidential election early next month.
Sunderland's intervention has been described as a "strong effort" by Turkish media, with the Black Cats understood to have put attractive personal terms to Brown. Their pitch centres on the guarantee of first-team football and a central role in manager Régis Le Bris's progressive system.
Brown is contracted at Fenerbahçe until June 2028, with an option for a further 12 months. A release clause of between €20 million and €23 million was reportedly included when he joined from Belgian club KAA Gent last July for around €8.5 million, though the Turkish club's current asking price sits below that threshold.
Atletico Madrid have also been credited with an interest, adding a third dimension to what is becoming a significant transfer battle.
Brown is an unusual profile at this level of the market. He came through the youth ranks at West Bromwich Albion and Derby County without making a senior appearance for either club before departing for Swiss side Lausanne-Sport as a free agent in 2021. He then moved to Gent and earned his Fenerbahçe move through consistent performances in Belgium.
This season, he contributed five goals and seven assists across 38 appearances, establishing himself as one of the more productive left-backs in the Turkish Süper Lig.
Fenerbahçe are in no immediate hurry. Brown has two years remaining on his deal and the club hold the option of a further year. If no offer meets their valuation, they retain a player who cost them a fraction of what is now being offered. The resolution, as with the club's other major decisions this summer, will follow the political calendar.