Randolph was said to be on the verge of a move to West Ham last week, but complications have delayed the deal.
The Irish stopper has still not been confirmed as a West Ham player.
And Woodgate is hoping for clarity over whether Middlesbrough will lose their goalkeeper or not this month in the next day.
“He’s still having the longest medical in history,” Woodgate said.
“He’ll have another scan and we’ll know more in the next 24 hours.
“I’ll get on with it, Darren’s a top pro, absolutely top class pro and if he goes to West Ham good luck and I wish him all the best.
“If he stays here, he’ll be fantastic.”
West Ham are thought to be fearful that Randolph isn’t fully over injury.
The Hammers need the 32-year-old to be fit, as they want him to cover for Lukasz Fabianski, who is currently on the treatment table as things stand.
Randolph would provide more reliable back-up at the London Stadium than Roberto has managed to this season, with the Spanish stopper being heavily criticised.
West Ham are next in action against Everton at the weekend, when Randolph could be in line to make his debut.
Have something to tell us about this article?