Teenager Mark O'Hara has signed a two-year extension with Kilmarnock.
Relegated Hull have released longest serving players Paul McShane and Liam Rosenior as they reshape their squad.
Spurs have been searching for a new centre-back following the departures of Steven Caulker and William Gallas.
The north London club failed in their bid to sign 23-year-old Chiriches last month, but a fee of around £8.5million appears to have now been agreed.
Steaua hope the defender, who has won 19 caps for Romania, will play in the second leg of their Champions League play-off against Legia Warsaw on Tuesday before completing his move to White Hart Lane.
"The two clubs have agreed to transfer him. I'm very happy for him," Steaua manager Laurentiu Reghecampf told a press conference on Saturday.
"I can not say the amount for which (he was) transferred, but it is very high.
"It will be the most expensive transfer in the history of Steaua.
"The owner says that all there is left to do is sign some documents.
"We hope he will help us Tuesday night if he can."
Chiriches has played at four different Romanian clubs and he also had a one-year spell at Benfica between 2007 and 2008.
An imposing figure at 6ft 1in, Chiriches can also play at full-back or in midfield.