It is understood that Lyons is on the final shortlist to take the £140,000-a-year post heading up the BBC Trust, which has been without a chairman since Grade's departure to become ITV's executive chairman.
Lyons currently works at the Office of the Prime Minster and is chairman of the English Cities Fund and the City Of Birmingham Orchestra. He has close links to Gordon Brown, the Chancellor of the Exchequer and expected next Prime Minister, which has led to accusations of cronyism.
He was previously chief executive of Wolverhampton, Nottingham and Birmingham city councils, where he was in charge of a £2bn budget.
Another contender for the role is believed to be Chris Powell, who chairs the Institute of Public Policy Research. Powell is the former chairman and chief executive of BMP DDB.
The Department of Culture Media and Sport, which has been involved in the decision-making process, said that the decision to appoint a new BBC chairman had not been finalised.
Meanwhile, it emerged in ITV's annual report this week that Grade was poached from the BBC to take up his new ITV role with the promise of more than £12m over a five-year period if he can turn around the fortunes of the struggling commercial broadcaster.
The deal means that Grade will be handed shares worth £6m as part of a five-year incentive plan published. The deal is dependent on the price of the broadcaster's shares in 2012.
Grade must make sure the group's revenues rise by 5% annually to receive the extra share dividend.
It was also revealed in the annual report that Charles Allen, Grade's predecessor, walked away from the broadcaster after earning £5.4m in his final year as the group's chief executive. The figure included £4.2m in compensation for his early departure in August last year.