The 29-year-old has agreed a move to The Riverside Stadium and was training with his new team-mates on Thursday.
Crossley left for Fulham earlier this week in a £500,00 deal, but manager Steve McClaren had already earmarked Nash as a replacement.
Nash was well down the pecking order at The City of Manchester Stadium, with David Seaman as first choice, leaving Nicky Weaver fighting with Danish youngster Kevin Ellegaard for a place on the bench - and so he has now accepted a move to Teesside.
The former Stockport County shot-stopper will arrive at Boro as number two, but will hope to push Australian star Mark Schwarzer for his number one jersey.
