According to BBC, Morgan and Pawar have struck a deal that will see the former replacing Ray Mali of South Africa for the next couple of years while the latter would assume the top job afterwards.
Mali became the acting president of cricket's governing body after Percy Sonn's death in May this year.
The ICC officials are currently in London for a series of meetings ahead of the council's annual conference in the Long Room at Lord's on Friday. A formal announcement of the Morgan-Pawar decision is expected later this week.
Welshman Morgan has been the chairman of the ECB since January 2003, having previously held a similar role with county side Glamorgan.
The impasse over whether he or Pawar should become ICC president has lasted since February when a nominations committee was unable to agree whose name to put forward.
It prompted a decision to extend then president Percy Sonn's term of office by a year until 2009.
But his death, the search for a compromise was stepped up with a view to a permanent replacement taking over next year.