The 32-year-old was last season’s Championship goalkeeper of the year, but injuries have restricted him to 14 league starts this term, the last of them at home to Hull City in November. He has been out with a thigh injury since, but that does not appear to have put off David Moyes, with a transfer expected to be finalised in the next 48 hours.
Republic of Ireland international Randolph made 28 Premier League appearances in his first spell with the Hammers before joining Boro in 2017.
After losing Adrian to Liverpool in the summer, West Ham signed Roberto to replace him, but already seem to have lost faith in the Spaniard.
Boro have had to scale back their budget now that they no longer receive parachute payments, and Randolph is one of their highest earners.
Aynsley Pears, son of former Boro goalkeeper Stephen, made his Championship debut in October, and has made another 11 appearances since.
Boro boss Jonathan Woodgate has told fringe players Marcus Browne and Marc Bola they can leave on loan this month.
“I spoke to Browne two days ago and said he’d be able to go out on loan, I think it’s important Marcus gets games so there will be a few teams in for him,” he said. “Likewise with Marc Bola, he’ll be able to go out on loan and come back in pre-season and fight for his place.
“I think it’s important that with their careers they don’t stagnate and they go out and play games, we’ll be watching them all the way through the season and monitoring their progress.”
Hull City coach Grant McCann has said the Tigers will “move on” if they are unable to complete the signing of Mallik Wilks this week.
Hull went into the transfer window with high hopes they could sign the former Leeds United winger, but the deal has stalled and with McCann eager to do business quickly, he is not prepared to wait around.
“We will wait over the next few days because if it doesn’t happen we will move on to the next one,“ he said. “People want over the odds and over the top at times and what we have gone in with is completely fair and is a good offer. If it’s a no we’ll move on, but I’m not going to wait about because we need another body in the attacking end of the pitch.”
Hull have already signed Liverpool midfielder Herbie Kane on loan.