The aim is to develop England's white-ball depth ahead of the country staging next year's 50-over World Cup.
The new national selector will be appointed by ECB before Test series with Pakistan in May.
Strauss had a chequered relationship with star batsman Pietersen after taking over the England captaincy from him and as an administrator effectively ended Pietersen's international career in 2015.
Morgan also spoke about the idea of having different coaches for different formats.
Strauss left for England after the second Test at Adelaide and was to return before start of Boxing-Day Test in Melbourne.
England head coach Trevor Bayliss was furious at Bairstow, calling his behaviour "dumb" on Monday, accusing him of handing Australia "ammunition".
Andrew Strauss has admitted the ongoing absence of Ben Stokes from The Ashes 2017-18 is a "blow" to the team.
The first Ashes Test takes place in Brisbane from November 23, with further matches to follow in Adelaide, Perth, Melbourne and Sydney.
Mark Stoneman will be Alastair Cook’s 12th opening partner since the retirement of Andrew Strauss in 2012.
England are scheduled to play two more day-night Tests in the upcoming season against New Zealand and Australia.