Randolph is the latest goalkeeper linked with a move to Middlesbrough, who have allowed Victor Valdes and Brad Guzan to leave this summer.
West Ham United's Darren Randolph
But Garry Monk's pursuit of a replacement has so far failed to pay dividends, with West Ham appearing to have followed the lead of Newcastle United and Sheffield Wednesday in issuing a negative response.
The Sun reported last week that Newcastle have turned down Boro's £5 million offer for Karl Darlow, while HITC Sport understands that Wednesday did likewise when faced with an undisclosed bid for Keiren Westwood last month.
West Ham were said to be in talks to sign a new 'keeper themselves in Joe Hart, though The Telegraph has since reported that the Hammers only want to loan the Manchester City outcast, with the Blues seeking a permanent transfer.
According to The Sun [12 July, page 51], Hart's arrival at the London Stadium is more likely to impact the future of West Ham number one Adrian than that of Randolph.
West Ham United's Adrian
The Mirror reports [12 July, page 49] that Middlesbrough are set to beat Newcastle to the signing of Liverpool winger Sheyi Ojo.