The Derbyshire chairman Chris Middleton has written to his 17 fellow county chairmen and the Test and County Cricket Board calling for them to push for Dexter's resignation before the board meeting on 17 August.
Dexter, whose position as chairman of the England committee comes up for renewal in March, cannot be sacked but he is likely to resign if he feels an overwhelming majority of chairmen wish him to.
In the four and a half years he has been in charge, the former England captain has been increasingly, in his own words, 'lampooned and harpooned'. But how responsible is Dexter for England's failures and should he follow Graham Gooch and stand aside? There is a case for Lord Ted. His role as chairman of selectors is his most public task, but his remit extends beyond that.
His job, for which he is paid pounds 30,000 a year, is part-time and deals with the whole range of English international cricket, from the under-15s to the senior side. Among his credits are the advent of the England observers in the county game, the increase in tours by junior teams and the streamlining of the youth structure.
He was also one of four-day cricket's prime movers, with the objective of increasing the practice and recuperation time available to players, while reducing their travelling.
But it will take time for these developments to pay dividends in the Test arena and his problem now, as Mike Smith, the chairman of Warwickshire, said yesterday, is that England's players 'have not performed to his or their own expectations'.
Smith, regarded as a possible replacement, added: 'Selection is a matter of opinion and most people would have picked the majority of the players used this season. I have considerable sympathy for Ted and am not one of those looking for his head.'
Given Dexter's limited role in selection - the team has been largely determined throughout his time in charge by the coach and captain - perhaps the best solution would be for him to become a non-voting chairman of selectors, which would enable him to continue his wider brief while taking away one of the main areas of criticism.Reuse content