He will fly home immediately from the three-week trip which also includes stops in Sydney, Japan and Hong Kong.
It represents a major blow to Rooney and to David Moyes, the new manager, who is intent on making an impressive start to his tenure, with the striker at risk of missing the beginning of the season.
There was already doubt that, with Rooney unsettled at United and refusing to confirm he wants to stay, the forward would start the campaign in United colours. The hamstring injury, sustained during training, further clouds his future.
"Gutted to have picked up a niggling injury especially as training was going so well, heading back to Manchester to have it assessed fully but early indications are nothing to worry about," Rooney posted on Facebook.
In a statement, United said: "Following a scan this evening on a hamstring injury sustained in training, it has been decided that Wayne Rooney should return home immediately for further assessment and rehabilitation. It is anticipated he will be unavailable for selection for a month."
The injury comes as a surprise after Moyes had stated on Wednesday that Rooney had returned for pre-season last week in his best physical form for many years. The Scot said: "They're telling me [at the club] they think it's the best shape they've seen him in for a long time."
Moyes also reiterated then that Rooney was not for sale though, with the 27-year-old refusing to publicly say he is content at United but the sense remains that, if the club received a serious bid in the region of £30m, they would consider it.
Now, with Rooney injured, it remains to be seen whether he will decide to make it clear he views his long-term future at United and focus on regaining fitness. Before the injury was sustained on Thursday, Rooney had appeared in good spirits, being photographed smiling alongside Patrice Evra, as Moyes prepared for Saturday's game with a Singha All Star XI, the first time the Liverpudlian would have played under the manager for United.
United play in the Community Shield on 11 August and start their Premier League season the following weekend. Rooney must be a major doubt for England's friendly against Scotland on 14 August.
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