The youngsters relation to Cole adds immediate pressure to succeed, though it appears the teenager is keen to follow in his father's footsteps and break onto the big stage.
Cole, currently on trial with West Ham, played in the development squad's 1-0 defeat to Colchester United on Monday night and appeared to impress Westley.
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"He is someone we will look at seriously. We think we are short in those forward positions," Westley said.
"Unfortunately tonight it was difficult for a front player, but he showed flashes that he is dangerous in behind."
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Recently-instated Hammers manager Slaven Bilic could view Cole as a cheap option as a backup striker, with his Javier Hernandez bid reportedly being rejected and Charlie Austin proving to be too expensive.
And while he would not be the first-choice pick, and perhaps nor the second or third, the 20-year-old scored five goals in 19 league appearances for Barnsley on loan last season and proved he has the potential to grow into a player worthy of the Premier League.
If Cole signs he will be the second player to join the club from City, following the earlier arrival of Norwegian teenager Martin Samuelson, who has already done well in pre-season and impressed the West Ham faithful.