Jennings was born in Scotland and as a child moved to London, England. He is the grandson of a former Clapton Orient player. Jennings attended University of Hull and later worked for the Sunday Times' Insight team in the late 1960s, after which he worked for other British newspapers before becoming an investigative reporter on BBC Radio Four's Checkpoint. In 1986 the BBC refused to broadcast his documentary concerning corruption in Scotland Yard; Jennings reacted by resigning and transforming the material into his first book, Scotland Yard's Cocaine Connection, and the documentary was aired by World in Action.
Jennings subsequently worked for Granada, filming several international investigations and small documentaries. His investigation of British involvement in the Iran-Contra affair won the gold medal at the 1989 New York TV Festival. In 1993, Jennings entered Chechnya with the first western TV crew ever to enter the country, to investigate Caucasus mafia activity. 1997 saw Jennings working with World in Action, with an investigation on British Olympic swimming coach Hamilton Bland, and in 1998 he presented a documentary on rail privatisation.
His first appearance on Panorama, a current affairs documentary programme, came in June 2006 (episode entitled "The Beautiful Bung: Corruption and the World Cup") when Jennings investigated several allegations of bribery within FIFA, including million-dollar bribes to secure marketing rights for the body's sports marketing company ISL along with vote-buying (to secure the position of FIFA president Sepp Blatter), bribery and graft attributed to CONCACAF president Jack Warner. It was followed up in October 2007 with an episode entitled "FIFA and Coe" exploring the relationship between former British Olympian Sebastian Coe and the FIFA Ethics Committee.
The most prominent programme was FIFA's Dirty Secrets (first aired on 29 November 2010) which was a 30-minute investigation of corruption allegations against some of the FIFA executive committee members who were to vote on the host for the 2018 FIFA World Cup. Jennings alleged that Ricardo Teixeira, President of Brazil's Football Federation (CBF) and of the 2014 World Cup Organising Committee, Nicolás Léoz of Paraguay, President of the South American Football Confederation (CONMEBOL), and Issa Hayatou from Cameroon, President of the (CAF) all accepted bribes from a television marketing firm.
In December 2015, he presented a summary of the investigations into FIFA entitled Fifa, Sepp Blatter and Me for BBC's Panorama.