The IPCC said the allegation was not part of Operation Sacristy, a probe into alleged corruption at the force.
Operation Sacristy, a separate criminal investigation into allegations against Mr Price, Mr Bonnard and others is also under way.
Mr Bonnard, who was suspended when he was arrested in 2011 as part of corruption inquiry Operation Sacristy, has denied any wrongdoing.
You can also meet the former monks of the monastery at the sacristy, where their skulls and bones are neatly stacked on shelves.
The senior officers were arrested in August 2011 as part of Operation Sacristy, a criminal investigation into allegations of corruption at Cleveland Police.
BBC: Sean Price and Derek Bonnard drop Operation Sacristy claim
Sean Price argued that publicity about the outcome could prejudice jurors in the event of a trial following the outcome of Operation Sacristy.
He remains on bail as Operation Sacristy - a criminal investigation into allegations of corruption at the now defunct Cleveland Police Authority - continues.
Prosecutors said Monday the former altar boy whom Avery admitted molesting in the church sacristy is scheduled to take the witness stand this week.
Mr Price remains on bail while Operation Sacristy, a wider criminal investigation relating to individuals with past and present associations to Cleveland Police Authority, continues.
Mr Price and his deputy Derek Bonnard were arrested in August 2011 as part of Operation Sacristy, a criminal investigation into allegations of corruption at the force.
The former chief constable remains on bail as part of Operation Sacristy, a corruption investigation relating to individuals with past and present associations with Cleveland Police Authority.
The opening of the sacristy is one of several projects through which today's 25-strong Orthodox Christian fraternity is attempting, with considerable subtlety, to solve an almost insoluble problem.
Seven were deferred pending the outcome of the wider criminal investigation by Operation Sacristy, a corruption investigation relating to individuals with past and present associations with Cleveland Police Authority.
The pair had announced their intention to sue National Crime Agency director Keith Bristow in his role as head of Operation Sacristy along with North Yorkshire, Warwickshire and West Yorkshire Police - the forces whose officers are involved in the investigation.
BBC: Sean Price and Derek Bonnard drop Operation Sacristy claim
The suspension - said by the force to be a neutral act - follows Mr Price's sacking last year, allegations against his deputy Derek Bonnard, who faces a misconduct hearing on 4 March, and the wide-ranging Operation Sacristy investigation into alleged corruption.
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