The man killed in the crash is reported to be from the United Kingdom, while RTE report the young boy is the son of one of the parachutists who had been on board.
Their bodies have been recovered within the past hour and are being brought to the Midland Regional Hospital in Tullamore for a post-mortem examination.
The light aircraft, a Cessna Caravan, was carrying 16 parachute jumpers from the club.
The plane crashed near Mount Lucas shortly before 2.30pm on Sunday afternoon, May 13.
The Clonbullogue facility is popular with skydivers.
The Tenerife air disaster of 1977 saw two 747 jumbo jets collide in mid-air.
Sixteen parachutists had made their jumps ten minutes earlier before the plane crashed on bogland with the man and boy on board.
Man and child injured following aircraft crash in Leinster
The injured are the pilot believed to be a British national and a seven-year-old boy on board, said the report.
"Thankfully, the 16 (parachutists) got out and our thoughts and prayers are with those in the plane".
Fellow Offaly councillor Noel Cribbin said: "This is a big shock to the area".
"These flights have been happening for the last 20 years and this is the first time I've heard of a plane going down".
This is the second aircraft crash to occur in Ireland in less than a fortnight. It's understood the crash site is located away from the road.
It was located a number of kilometres from the airfield in bogland and a rescue and recovery operation is now underway.
"Then I heard a awful roaring and it went straight down".
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