His widow Daphne phoned the PM's office on Saturday to "kindly decline" the offer, Mr Abbott said.
The commentator himself had wished for "something very, very quiet and very, very private", he added.
Benaud, 84, is said to have died peacefully in his sleep on Thursday.
He will be remembered in a private funeral attended only by his immediate family, Australian media reported.
Mr Abbott told reporters on Sunday: "I thought it was important that as a mark of the respect that we have long had for him that we should offer a state funeral.
"But my understanding is that Richie's own wishes were for something very, very quiet, and something very, very private."
A pioneering leg-spin bowler, Benaud played in 63 Tests, 28 as captain, before retiring in 1964 to pursue a career in journalism and broadcasting including a long association with the BBC.
His final commentary in England came during the 2005 Ashes series, but he continued to work for Channel Nine in Australia until 2013.
Cricket Australia chairman Wally Edwards described Benaud as "the iconic voice of our summer".
The family of the late former Australian cricket captain and legendary cricket commentator Richie Benaud has rejected Australian PM Tony Abbott's proposal for a state funeral.