Photos of the graffiti show the recurring vandalism reading “NF” and “aliens out”.
Police have said that they are treating the graffiti as “extremely serious” and urge anyone with information to come forward.
Greenwich Council has said it is “appalled” by this graffiti appearing in the borough.
The reward is being offered by Crimestoppers, a charity working with police and Greenwich Council to eradicate this vandalism.
Sergeant Kristina Money, from the Neighbourhoods Policing Team in Eltham, said: “We are aware of the ongoing spate of racial graffiti across the Eltham area and are taking this extremely serious.
“We urge anyone with any information to come forward, and to continue to report any graffiti you may find.
“Should you be affected by any of these incidents please don’t hesitate to contact us so we can offer support.”
Police ask that you contact them by calling 101 and asking to speak to the Neighbourhoods Policing Team for Eltham.
A spokesperson for Greenwich Council said: “We are appalled by the vile, racist graffiti in Eltham.
“An investigation is ongoing and we’re working closely with the Metropolitan Police to bring the perpetrator to justice.
“This is something we take extremely seriously, which is why we have been removing the graffiti immediately.
“We’re also working with the charity Crimestoppers to offer a cash reward of up to £1,000 for anyone that comes forward with useful information.”
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