The word “F***” was scrawled over the huge artwork of the England footballer, painted on the side of a cafe in Copson Street in Withington, south Manchester.
The words “shit” and “bastard” were also graffitied beside the word “Saka”, the 19-year-old player whose final penalty miss gave Italy victory.
Rashford, Bukayo Saka and Jadon Sancho, who also missed a penalty, were all targeted with racist abuse on social media after the game.
Greater Manchester Police (GMP) said they were called at 2.50am on Monday to reports of the racially aggravated damage to the mural.
Chief Superintendent Paul Savill said: “This is disgraceful behaviour and will absolutely not be tolerated.
“Greater Manchester prides itself on being made up from a number of diverse communities, and hate crime in any form is completely unacceptable and not welcome here in our city.
“GMP takes crimes of this nature very seriously and an investigation has been launched.
“If anyone has any information that could help us to identify this offender, please do not hesitate to speak to police.”
The graffiti has since been covered over and locals have written messages of support and posted love hearts reading “hero” over the mural.
Manchester United Football Club has also tweeted a video of the notes left by fans and called Rashford an “inspiration”.
Rashford thanked fans for their support, tweeting: “The messages I have received today have been positively overwhelming and seeing the response in Withington had me on the verge of tears.
“The communities that always wrapped their arms around me continue to hold me up. I’m Marcus Rashford, 23 years old, black man from Withington and Wythenshawe, South Manchester.
“If I have nothing else, I have that. For all the kind messages, thank you. I’ll be back stronger. We’ll be back stronger.”
The mural, based on a photograph by Daniel Cheetham and painted by street artist Akse, was created last November in collaboration with Withington Walls, a community street art project.
It was reportedly commissioned in recognition of the Manchester United player’s work to tackle child food poverty.
Rashford’s mother provided the quote on the mural, which reads: “Take pride in knowing that your struggle will play the biggest role in your purpose.”
A crowdfunder set up to raise funds to repair the mural and keep maintaining it raised more than £20,000 in just a few hours.
It says: “The England team may have lost, but they have done us proud on and off the pitch. This team has shown us the nation we can be. They have proved that diversity is our strength.
“We are unsure how much it is going to cost at this stage to repair, but anything raised beyond what it costs will likely be spent on funding CCTV coverage, protective paint or further artworks in the village. Any funding raised beyond our needs will be donated to anti-racism and food bank projects.”
Prime Minister Boris Johnson and England manager Gareth Southgate have led the condemnation of those racially abusing black England players.
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