The Ligue 1 side are keen to spread the cost of the move and are offering a loan with obligation to buy next summer for £13.5m.
Rowe, who has also been linked with a move to Leeds, made 32 appearances for the Canaries last season, scoring 12 goals.
The 21-year-old's performances during the early part of last campaign saw him named the EFL Young Player of the Month for August.
The Norwich academy graduate was awarded for his form with a call-up to England u21s by Lee Carsley, and netted on his debut for the Young Lions in a 9-1 win against Serbia.
Marseille got off to a winning start in Ligue 1, with Mason Greenwood netting a brace at Brest
Marseille have made a number of new additions under De Zerbi in the transfer window, as they look to rebound from a disappointing 2023-24 season that saw them miss out on European football.
Canadian midfielder Ismael Kone has arrived from Watford, while Pierre-Emile Hojbjerg arrived on a season-long loan from Spurs last month.
Bamo Meite, Derek Cornelius, Jeffrey de Lange, Geronimo Rulli and Elye Wahi have all arrived on permanent deals this summer.
The club's marquee signing has been that of former England international Mason Greenwood after the 22-year-old completed a £27m switch from Manchester United.
Greenwood has enjoyed an immediate impact since arriving in the south of France and netted twice as they recorded a 5-1 win in their season opener against Brest on Saturday.