Sky News has learnt that Mr Griffith-Jones, who steps down as the Financial Conduct Authority (FCA) chairman next month, is among a handful of candidates to succeed Lord Currie at the CMA.
His inclusion on a shortlist submitted to the Department for Business, Energy and Industrial Strategy (BEIS) has surprised some in Whitehall.
Mr Griffith-Jones spent 37 years at KPMG, where he rose to became joint chairman of its European business.
KPMG and the other “big four” accountancy firms have been left to fight calls for their perceived oligopoly over the audit market for big companies.
Those demands have been sparked by the collapse of Carillion, which KPMG audited, and disclosures made to MPs this week showing that the quartet of firms earned £70m in fees from Carillion during the last decade.
Mr Griffith-Jones’ five-year stint as chair of the FCA and Payment Systems Regulator has also been chequered, with the City watchdog currently embroiled in a row with the Treasury Select Committee (TSC) over the treatment of small businesses by Royal Bank of Scotland.
He also faced criticism over the FCA’s handling of a crisis over a botched media briefing in 2014 which triggered a crash in the share prices of major insurance companies.
The search for a new chairman of the CMA comes as the competition watchdog anticipates a post-Brexit increase in its workload.
Sky News revealed in December that Andrew Tyrie, the former Conservative MP who chaired the TSC, was the preferred candidate of Greg Clark, the Business Secretary, to chair the CMA.
Mr Tyrie would be an intriguing but possibly contentious choice for the job, which pays a salary of up to £160,000, according to a Government recruitment advertisement.
The former Tory MP for Chichester, who stepped down at this year’s General Election, could be seen as an explicitly political choice for a role which is supposed to be neutral.
Mr Clark is said to be keen on seeing Mr Tyrie in the role amid a continued perception of dissatisfaction in ministerial ranks about the regulator’s handling of key probes into the banking and energy markets.
The CMA is facing a demanding period in the run-up to, and aftermath of, the UK’s departure from the European Union because of an increase in its post-Brexit remit.
The UK watchdog will be required to adjudicate on mergers which are presently under the jurisdiction of the European Commission, as well as many of the corporate competition issues currently overseen by Brussels.
In his resignation statement last September, Lord Currie alluded to this, saying: “Brexit is likely to generate a big increase in the CMA’s workload and its role in the world, and I owe it to my successor to allow them enough time to be involved in shaping its plans to respond to that challenge, and then take the organisation into the next phase of its development.”
Current cases consuming the watchdog including 21st Century Fox’s bid to acquire Sky plc, the owner of Sky News, and a number of enforcement actions relating to NHS drugs suppliers.
Other sector-wide inquiries include a consumer law investigation into hotel booking sites announced in October.
Interviews to identify Lord Currie’s successor are expected to take place during the coming weeks, according to a Whitehall source.
The FCA declined to comment on Mr Griffith-Jones’s candidacy.