Treasury officials and staff from the Shadow Chancellor's office have already held discussions in the search for a successor since Sir Andrew made it clear that he no longer wanted to stay on at SIB.
Sir Andrew's stated wish to step down at the end of his five-year term, notified to the Treasury two weeks ago, was immediately relayed to Labour.
However, it is understood that an initial Treasury bid to bring the search to a speedy conclusion was slowed down by the Labour team, who are prepared to delay an appointment until after the general election if necessary.
A spokesman for Mr Brown said: "We were disappointed that Sir Andrew did not want to carry on. He had a good understanding of where we wanted to go in the regulatory field.
"There have been discussions with civil servants. It has been 100 per cent clear from the outset that they have no intention of appointing someone who did not meet with Gordon Brown's approval."
The cross-party consensus has the full support of Mr Clarke and also Eddie George, Governor of the Bank of England, who will jointly appoint a successor to Sir Andrew.
The spokesman added: "But this is a very important appointment for us. We may not be able to find someone quickly, or in advance of the election."
If a replacement is not found in time, Sir Andrew may be asked to stay on beyond his departure date of 31 May, or the post will be left vacant.
A Treasury spokeswoman said the search for a successor was unlikely to be over quickly. Potential candidates would be considered carefully.
"We are searching for someone who has Sir Andrew's combination of interpersonal skills, industry experience and dynamism. That will take some time."
It is understood that no shortlist has yet been drawn up for consideration by Treasury Ministers or the shadow chancellor. One option being considered is to recruit an outside figure with sufficient senior executive experience and credibility for the job.
Among those most strongly tipped by sources yesterday were Christopher Sharples, former chairman at the Securities and Futures Authority. He is chairman of Datastream, and sits on the board of the GNI, the derivatives trading arm of the Gerrard Group, the stockbroker and fund management company.
He said: "Sir Andrew will be a tremendously hard act to follow. It is a tremendous challenge and I have have always been open to new challenges But I do have my hands very full at present."
Other candidates include Phillip Thorpe, chief executive at Imro, the fund management watchdog, who is credited with retrieving the regulator's credibility in the wake of the Maxwell pensions scandal.
An outside tip is Colette Bowe, chief executive at the Personal Investment Authority, who has previously held a senior position at SIB for several years. Other potential outside candidates include Nick Durlacher, the current SFA chairman who replaced Mr Sharples.