The Northern Irishman, who has also served as youth coach and assistant manager at the Gunners, is said to have been "in and out of hospital" following the diagnosis.
"I'm aware he has been in and out of hospital receiving treatment," a source told The Mirror.
Rice dedicated 44 years to Arsenal, turning out more than 500 times for the London outfit during his playing career, and serving as caretaker manager prior to the arrival of Arsene Wenger.
"Our thoughts are with Pat and his family. Everyone at Arsenal sends him our best wishes," an Arsenal spokesperson said in a statement.
Rice retired from football in May last year, and received an MBE for services to football in the 2013 New Year's Honours list.