Bowen, 29, is widely expected to leave West Ham following their relegation from the Premier League, five-and-a-half years after joining from Hull.
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The winger has been arguably their best player throughout his time at the club, becoming an England international and scoring 85 goals in his 280 games.
According to journalist Miguel Delaney, Chelsea are now weighing a move for Bowen as they seek to add more Premier League experience to their squad.
It’s understood that West Ham will demand a fee of around £50 million but clubs believe a deal could be done for closer to £40m.
Chelsea aren’t the only club interested, however, with a number of other top flight clubs keen on a move for Bowen.