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The 22-year-old striker faces at least six weeks on the sidelines, with United boss Ole Gunnar Solskjaer confirming the full extent of the injury after Sunday's 2-0 Premier League defeat at Liverpool.
Rashford, who has scored 22 goals for club and country this season, tweeted: "Blink and I'll be back, fitter than ever. See you soon."
The England international also said he would remain heavily involved in all aspects of club life at Old Trafford during his rehabilitation.
"On the pitch or not, @ManUtd fans don't get much bigger than me," he added.
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"You best believe I'll be involved in team meetings, team huddles and team sessions throughout this recovery."
Rashford sustained the injury after being introduced as a second-half substitute in last week's FA Cup third-round replay win over Wolves.
Speaking after Sunday's defeat at Anfield, Solskjaer told Sky Sports: "He's suffered a bad injury, it's a stress fracture.
"It happened against Wolverhampton and it is not something that has happened before. I'm looking at Marcus to be out for a while. He won't be back, I'm not a doctor so I don't know exactly, but we are looking at several weeks and then some rehab after that probably.
"We've had many injuries to big players this season, it's just an unfortunate position that we are in. The window is open and it might be that we look at something for the short term."
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