According to a report in the Sunday Times, Dyke, who was forced to quit as director-general in January following the publication of the Hutton Report, is being urged by close friends to apply to Grade and make a welcome return to the BBC.
Dyke has said since he left that he did not want to go and that he had only tendered his resignation because he thought the BBC board of governors would support him and decline to accept it.
His departure saw a massive outpouring of support for Dyke by BBC staff, who held the director-general in high regard.
Appointing a director-general will be one of Grade's first challenges when he takes up the job of chairman on May 17.
The Sunday Times says that Grade has told friends that Dyke, who is an old friend and former colleague of the new chairman, could be a contender again for the director-general's job.
It could be a return that is fraught with controversy after the bruising battle with the government over the Hutton Report that followed the death of MoD scientist Dr David Kelly.
However, those close to Dyke say this need not necessarily be the case, arguing that his reappointment would not rankle Number 10 because Tony Blair is said never to have wanted Dyke to lose his job.
If Dyke and Grade were reunited it would be a trip back in time for the pair to when they worked together in the 1970s when Grade was director of programmes at London Weekend Television and Dyke was editor of 'The Six o'Clock Show'.
If Dyke does run for the job he will possibly be up against Channel 4 chief executive Mark Thompson although it is not clear if, after only two years at the station, it is too soon for the former head of BBC One to make his return.
Also in the running for the job are BBC director of radio Jenny Abramsky and Jana Bennett, the BBC's head of television who is being tipped as the first woman to lead the corporation.
With the appointment of Grade the acting director-general Mark Byford, who has the support of Number 10 policy adviser and former director-general John Birt, is not believed to have much chance.
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