The 10ft-high bronze statue was unveiled on 4 November – a week before the centenary of the Armistice – but less than week later had been sprayed with the words “Sepoys no more”.
The memorial was commissioned by Sikh temple Guru Nanak Gurdwara Smethwick to mark the sacrifices made by south Asian servicemen of all faiths.
The term “sepoy” refers to Indian soldiers serving in the British or other European armies.
The words “of the Great War” from the statue’s title had been sprayed with a black line, while the words “1 jarnoil” had been graffitied on to the memorial.
There was speculation on social media that the phrase could refer to Jarnail Singh Bhindranwale – a controversial figure in Indian history considered by some to be a terrorist who wanted to create a Sikh state.
He was killed in 1984 in a gun battle with the Indian army after his organisation occupied the famous Golden Temple complex in the city of Amritsar.
West Midlands Police said it is treating the incident as aggravated criminal damage.
The force said CCTV is being recovered and officers are working with worshippers and management at the temple.
Sergeant Bill Gill, from the Smethwick neighbourhood team, said: “We understand that this attack has caused a lot of concern in the community, and we are working to understand the reasons behind it and identify whoever is responsible.
“Officers had already planned to be at the remembrance event which is happening tomorrow at the statue. I’d urge anyone with concerns to speak to the officers attending the event.”
Anyone with any information is urged to contact West Midlands Police or call Crimestoppers anonymously on 0800 555 111.