Rose has not been allotted a squad number at Tottenham this season and is firmly out of Jose Mourinho’s plans going forward in north London.
The defender failed to find a club through the international transfer window, but it has been speculated he could drop down to the Championship before the domestic window shuts, in search of game time.
Middlesbrough have been linked with potentially offering Rose a home in the north east this season.
A switch north was the course Rose took in January, when he headed to Newcastle United on loan.
But it has been claimed that Boro will not be making a move to sign the England full-back before Friday’s transfer deadline.
Middlesbrough are hopeful of making additions before the window closes, but Rose will not be arriving.
Rose’s wage demands are too high for Middlesbrough and the Championship club do not want to pay over the odds for the 30-year-old full-back.
Boro are in the market for a left-back and have held talks to sign Chelsea’s Baba Rahman on loan.