The news comes just two days after manager Tony Pulis left the Premier League club.
Nzonzi, 24, only joined from Blackburn last summer, but has already set his sights on a move away from the Britannia.
Stoke chairman Peter Coates told the Daily Telegraph: "I can confirm that we have received a written transfer request from Steven Nzonzi.
"We are not unduly worried about the situation as he is our player and has three years remaining on his contract."
Pulis departed the club on Tuesday after seven years in charge.
Stoke have not yet appointed a replacement but whoever it is will surely want Nzonzi, who was named the club's young player of the year, on board.
Meanwhile, Begovic has hinted he will be joining Liverpool this summer, according to reports.
He is quoted in The Sun saying: "I want to start preparing on July 10, with all the players of Liverpool."
The highly-rated goalkeeper said earlier this season he expected to be sold after Stoke signed England goalkeeper Jack Butland in the January transfer window.