The Everton star is yet to feature for England at Euro 2016.
HITC Sport understands that Boro are hoping to finalise deal for Marten De Roon this week.
Aston Villa are reportedly keen on Watford shot stopper Romelu Lukaku.
The two players secured moves to Ibrox earlier this month.
Walters was a second-half substitute during Ireland's 2-1 defeat to France on Sunday, despite admitting he was not fully fit.
The West Ham United star has been the key man for France at Euro 2016.
The Leeds winger played a big role for Northern Ireland but suffered heartbreak in the last 16.
Glasgow Rangers manager Mark Warburton wants his side to demand the maximum from each other.
The young talent has been excellent for England so far this summer.
Leeds United target Ashley Fletcher will be joining Manchester United on their pre-season tour to China next month.