They begin a game and then start asking where the women live and whether they are married and want to continue chatting via other messaging apps such as WhatsApp.
It is likely many of them are romance scammers.
Scopely said the chat function could be restricted to friends only.
One woman, who is in her 60s and lives in London, told BBC News she was contacted via private message by two or three people per week, all claiming to be men from the US.
She did not wish to be named.
"It's almost like a script," she said.
"They start with, 'How you doing?' They match you to start a game, then start messaging.
"They play very badly, so you win the game. And then they big you up.
"Regularly, they say, 'I just want to check, can't we be friends?'
"When you say, 'No,' some of them disappear, they resign from the game.
"If you don't reply at all, most of them resign from the game."
She believes many of them are scammers.
English did not appear to be their first language, she said.
One man confessed to using his son's photograph as a profile picture, because he thought his son was "more attractive".
"This is not a dating site," she said.
Scopely said it "does not tolerate any harassment or misconduct" on its games platforms and players should report incidents to it.
"In Scrabble Go, players are able to access mute and block functions within the chat feature, as well as the 'mute public chat' privacy setting," a spokeswoman said.
"When enabled, players will only receive chat notifications and messages from players they already know and are connected with as a Facebook friend, favourite, or via their synced contacts."