In 2014, climate sceptics accused the BoM of manipulating temperature data to accord with global warming predictions.
There was no independent evidence to back the claims.
Accurate temperature records play a key role in climate change modelling and forecasts.
Documents released to the Australian Broadcasting Corp (ABC) via a Freedom of Information request, show that Mr Abbott's Department of Prime Minister and Cabinet wanted to conduct due diligence on the Bureau as part of a broader task force that was providing advice on post 2020 carbon emissions reduction targets.
The allegations of data manipulation were originally published by News Corp newspaper The Australian, in August 2014, based on comments made by climate change sceptic and biologist Jennifer Marohasy.
"In (George Orwell's) Nineteen Eighty-Four, Winston Smith knows that, 'He who controls the present controls the past'. Certainly the Bureau appears intent on proving the historical temperature record by changing it," Ms Marohasy said, The Australian reported.
The Bureau denied altering climate records to exaggerate estimates of global warming.
"Our role is to make meteorological measurements, and to curate, analyse and communicate the data for use in decision making and to support public understanding," it said at the time.
Environment Minister Greg Hunt defended the Bureau, arguing against an investigation.
"It is important to note that public trust in the Bureau's data and forecasts, particularly as they relate to bushfires and cyclones, is paramount," Mr Hunt wrote to Mr Abbott.
After winning government in 2013, Mr Abbott scrapped a number of initiatives introduced by previous governments to address rising levels of carbon dioxide in the atmosphere.
The probe was eventually dropped.
Scientists and climate change activists are hoping Australia's new Prime Minister, Malcolm Turnbull, who has previously called for tough action against climate change, will change the government's current policies.
Former Australian PM Tony Abbott's office wanted to investigate the Bureau of Meteorology (BoM) amid claims it had exaggerated climate change forecasts, documents show.