The 24-year-old is out-of-contract in the summer and has thus far shunned any attempts to keep him at the club.
This is mainly due to the fact Gareth Southgate dropped him earlier this year in favour of Brad Jones and Turnbull has been warming the bench ever since.
Skysports.com understands that Chelsea and Newcastle are leading the chase, although a number of other clubs - including Scottish champions Celtic - are also hot on his trail.
Chelsea are keen to bolster their squad with young English talent and Turnbull fits that bill and would provide excellent back-up to number one Petr Cech - something they are short of since selling Carlo Cudicini in January.
Newcastle's interest comes through their respected goalkeeping coach Paul Barron - who spent a number of years at Middlesbrough before moving to St James' Park and he played a major part in Turnbull's early education in the game.
Turnbull for his part insists he is fully committed to Boro's battle against relegation and will look at his own situation in the summer.
"We've said we're going to wait until the end of the season now," he said.
"There's only a month to go, so this is not the time to be talking about contracts or anything like that.
"The most important thing is the games that are coming up - everything else can wait until the summer."