Rhiannon Spear, Greater Pollok representative, tweeted after UK hopeful James Newman scored nil points in the Eurovision final, won by Italy.
The SNP's national women's convenor posted: “It's ok Europe we hate the United Kingdom too. Love, Scotland.”
She later apologised for the tweet and deleted her Twitter account.
Her post was branded by the Scottish Conservatives as another example of the SNP’s “toxic obsession with division”.
A spokeswoman for Police Scotland said complaints abut the tweet were being investigated and assessed.
She said: “We have received a number of complaints regarding a tweet posted on Saturday, May 22 and these are currently being assessed.”
Scottish Conservative chief whip Stephen Kerr added: “This is abhorrent language from one of the SNP’s most prominent campaigners and a 2021 election candidate, not to mention chair of Glasgow City Council’s education committee. We should be teaching tolerance and inclusion, not hate and division.”
In an apology, Ms Spear said: “I have now deleted this tweet about the UK's results in the Eurovision Song Contest, and apologise for any offence caused.”
UK entry Mr Newman, who is the older brother of pop star John Newman, had hoped to win over viewers with his rendition of upbeat track Embers, which was inspired by the end of lockdown.
He performed in front of an arena of 3,500 fans in Rotterdam, in the Netherlands.