"Manager Tony Pulis has been looking to strengthen the squad with the intention of taking us to an even higher level after we achieved 12th place in the Premier League last season.
"We are confident Robert is the type of player who can help us achieve that aim."
Huth made his English football debut with Chelsea in 2002 at 17 years old but only played 41 league games in four years before leaving for Middlesbrough in a £6m deal. He has 19 international caps.
If the fee for Huth does go up to £6m, it will eclipse the club record £5.5m Stoke paid Reading for striker Dave Kitson in June 2008. The fee for Tuncay is expected to be similar.
Tuncay has scored 15 goals in 58 appearances for Turkey, and finished as Boro's top scorer in the 2008-09 season, netting eight times in 33 starts.
Robert came over here early in his career and did well at Chelsea and Middlesbrough. We know all about him, he has plenty of experience in the Premier League
Stoke assistant boss Peter Reid
Stoke hope to have both players registered in time for Saturday's game against Sunderland.
Middlesbrough boss Gareth Southgate had been keen to keep the nucleus of his squad together, following their relegation from the Premier League in May.
England midfielder Stewart Downing, Egyptian striker Mido and goalkeeper Ross Turnbull were the only high-profile names to leave the Riverside during the close season.
Stoke have been desperate to strengthen their squad before the transfer deadline, with midfielder Dean Whitehead from Sunderland their only major arrival during the summer.
Stoke assistant manager Peter Reid told BBC Radio Stoke: "Robert came over here early in his career and did well at Chelsea and Middlesbrough. We know all about him, he has plenty of experience in the Premier League."
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