Morgan presided over the nation’s maiden World Cup triumph following a dramatic victory over New Zealand at Lord’s on Sunday.
The 32-year-old, however, is considering his future as skipper knowing he is unlikely to be playing when the next global showpiece arrives in 2023.
With the T20 World Cup taking place next year, former England captain Strauss believes Morgan’s influential role in shaping a golden era of one-day cricket entitles him to choose the timing of his departure.
“Eoin has earned the right to do it,” said Strauss. “The question for him is what he wants to achieve here because he has climbed Everest.
“That is the question for all the players because we have made mistakes in the past.
“We’ve won Ashes series and got to No 1 in the world and thought that was the end in itself. We have to find a way of making this a launchpad for something bigger.
“I certainly hope that what Eoin is doing right now – unless he is clear he wants to carry on – is just taking a bit of time to reflect where he is at.
“To stay on as captain he needs to be driven and motivated to push people on, as he has done over the last four years. If he has the bit between his teeth then we’ve seen just what a fantastic leader he is.
“The rest of the team will follow him until the last day he is on a cricket field, and he needs to decide when that is.”
Acting in the capacity of director of England cricket, it was Strauss’s decision to retain Morgan as captain in the wake of a dismal World Cup group exit in 2015.
In reward, Morgan has overseen a radical transformation, with England now a swashbuckling side that has swept all before them.
“After watching the last World Cup, I was genuinely angry by what I’d seen,” Strauss said.
“We were playing the wrong brand of cricket, picking Test players and hoping they’d put some of the best one-day players in the world under pressure.
“(We were) Playing a conservative brand of cricket and trying to justify the unjustifiable. I still remember my quote from the 2007 World Cup that two threes are as good as a six! To me, and this was before I was appointed the director of cricket, we just could not do this again.”
Strauss felt it was imperative that going into the next World Cup that England prioritised one-day cricket, and with a style “that genuinely pressurised the opposition”.
“Once I was appointed director of cricket, well, then it was a case of ‘If we’re going to prioritise one-day cricket what do we need?’” questioned Strass.
“If we want to play that type of style, who is the right captain to embody that? That led me very quickly to Eoin Morgan.”
Lord’s will be turning red for day two of the second Ashes Test on August 15 in aid of the Ruth Strauss Foundation, the charity set up by Strauss after his wife died of lung cancer in December 2018.
The foundation provides grants for research into rare forms of lung cancers and provides emotional and wellbeing support to patients and their families.
“One thing that has just blown me away, effectively from the day Ruth died, was the incredible wave of support I’ve received from the cricket family and fraternity, all around the world,” Strauss said. “This is the embodiment of that really. Doors have been opened that wouldn’t usually be opened, everyone just said ‘Let’s do it, let’s commit and make it a special day’.”