The France international has a £35 million buy-out clause inserted into his contract with Sevilla, but the La Liga side could be tempted to accept just £22 million for his services, according to the Daily Mail.
The 29-year-old is an aggressive, tenacious central midfield player, and the quality that he possesses undoubtedly makes him an ideal addition to either Arsenal or Everton's respective squads.
He also has experience of playing in the Premier League, having spent time with Blackburn Rovers and Stoke City.
The Daily Mail claims that he is desperate to leave the Ramón Sánchez Pizjuán Stadium this summer, and is prepared to force through a move by handing in a transfer request.
The Daily Mail claimed that both the Gunners and the Toffees held talks with Sevilla regarding the possibility of signing N'Zonzi back in December 2017, but a move to England obviously failed to materialise for the Frenchman.
Following the latest developments, though, both Arsenal and Everton could very well be tempted to renew their interest in the 6ft 5in powerhouse.