The 24-year-old Frenchman, whose current contract with the Ligue 1 outfit runs out in June 2017, has scored 10 goals and provided two assists in 27 league matches.
"England, this is my dream," he told L'Equipe. "Since childhood I've dreamed of Arsenal.
"The main thing is to believe and never give up. Here I am in Monaco, I have three years of contract.
"There will surely be reinforcements [this summer]. We'll see what happens."
Riviere joined Monaco from Toulouse in January 2013.