As reported last week, Boro were closing in on the 24-year-old Namibia international after his contract at Ewood Park came to an end.
Chris Wilder thought he had secured a deal for Nyambe, however sudden interest from overseas parties has complicated things.
The unnamed French clubs entering the race just as Boro were hoping to push a deal over the line has made Nyambe delay a decision on his future, although the Teessiders are still keen to sign the right-back despite the complications.
Nyambe has been a regular for Blackburn since his emergence into the senior squad as a teenager, and last season made 31 appearances in the Championship, having missed a chunk of the campaign due to injuries.
He now has a decision to weigh up as he can either take the offer from Boro on the table or take his chance abroad.