The 30-year-old slipped down the pecking order at the London Stadium following Joe Hart's arrival from Manchester City and with the Irish international keen to secure regular game time ahead of next year's World Cup, he looks set to move to the ambitious Teessiders.
'The Daily Mail' reports that the two clubs are in advanced talks over the former Birmingham man and that a deal could be tied up in the coming days.
Randolph made 22 Premier League appearances for the 'Hammers' last season as they endured a frustrating campaign.
Boro boss Monk has been eager to sign a new keeper this summer following the departures of Brad Guzan and Victor Valdes.
Randolph's stock in the Championship is high, following a spell with Brimingham City earlier in his career and the signing will come as a further boost to Boro fans as the club aims to return to the promised land of the Premier League at the first time of asking.