While the England and Wales Cricket Board (ECB) have yet to confirm Cook's departure as ODI skipper - which is expected to be announced on Saturday - Pietersen added credence to the speculation on Twitter.
Cook's tenure as ODI skipper has come under increasing pressure during England's 5-2 series defeat in Sri Lanka over the past month.
Pietersen is currently in Australia playing for the Melbourne Stars in the Big Bash.
"Great news that @Eoin16 [Morgan] is new England 1-day captain," Pietersen wrote on Twitter.
"Top player, leader and bloke. Hope I get to play for him."
Cook has made five centuries and 19 half-centuries in 92 ODI matches but his form with the bat has deteriorated badly.
The 29-year-old has reached 50 just once in his past 22 ODI innings and calls for him to resign have gathered pace after England lost their last three 50-over series against Test-playing nations.
With the Cricket World Cup in Australia and New Zealand set to get under way in February, Cook has regularly insisted he will not walk away from the role, stating his determination to improve his own and the team's form.
However, with the announcement of England's provisional 30-man squad to come on Saturday, widespread reports on Friday suggested that the decision had been taken out of his hands by the selectors.
Former England captain Michael Vaughan applauded the prospective move on Twitter: "Congrats @Eoin16 .... The right man for the job....."
The timing of the move - should it be confirmed - may raise some eyebrows so soon ahead of the World Cup, with Alex Hales now possibly set for an extended run in the side at his preferred position at the top of the order.