Speaking at a Policing Board meeting, Mr Baggott said he will be writing to former Detective Chief Superintendent Norman Baxter, who claimed Downing Street had tried to pervert the course of justice by asking for the release of republican suspects.
"It was a serious allegation made in the context of parliamentary privilege, but in relation to that now we want to establish exactly what the facts are," he commented.
He added that he stood by his apology following the collapse of the case against John Downey who was charged in connection with the Hyde Park bombing in 1982.
It brought to light the "comfort letters" which were sent to 187 on-the-run republicans.
Mr Downey was mistakenly informed in 2007 that there was no interest in him from them, or any other police force across the UK.
Judge Mr Justice Sweeney blamed the blunder on the PSNI, describing it as a "catastrophic error".
At the time the Chief Constable said that the force accepts "full responsibility" for the failures and referred the matter to the Police Ombudsman.
"I am clear that the apology I gave in the aftermath of this judgement was appropriate, I stand by that, it was appropriate for the victims and victims' family [members]," he said on Thursday.
"It was appropriate in terms of respecting a learned judge-led inquiry and I believe it's appropriate in terms of facts of the period in that judgement and chronology, which are very detailed and very specific."
He told the Policing board that he was aware of the OTR review, but said he would reserve his opinion on whether it was managed properly and ethically for his Northern Ireland Affairs Select Committee appearance.
It is wrong that the Northern Ireland Policing Board should be treated in this manner. If the Chief Constable is prepared to give answers to a Westminster committee but not to the Policing Board, it undermines the scrutiny role of the board.
Jonathan Craig, DUP
DUP Policing Board member Jonathan Craig said he was deeply dissatisfied by Mr Baggott's answers at the meeting.
"For the second month in a row, the Chief Constable sought to stone-wall the Northern Ireland Policing Board concerning the so-called 'On The Runs' administrative scheme," the Lagan Valley MLA said.
"He pointedly refused to answer several questions that I put to him, relying upon a defence that asked people to wait for the conclusion of various inquiries."
Meanwhile the victim of the republican murder attempt Mr Downey referred to on Wednesday has spoken of his "anger" and "disgust" following the allegations.
Mr Brush, now a DUP councillor in Dungannon, told UTV on Thursday that he was "surprised but not shocked" by the revelations.
"It makes me feel disgusted and angry that such a thing should happen. I'm just wondering how many other phone calls of a similar kind have happened in other cases," he told UTV.
"And how many people are now walking free because of the interference of the Northern Ireland Office or Downing Street in the cases."
I'm just thankful that Norman Baxter and the people in charge of that case were strong enough with their enquiries and did not allow political interference to stop them.
Mr Baxter continued the investigation which ended in the conviction of Gerry McGeough for attempted murder.
Mr Adams has said he protested to the British Government, but did not ask them to intervene.
At the Northern Ireland Affairs Select Committee meeting on Wednesday, both Mr Baxter and former Assistant Chief Constable Peter Sheridan denied any knowledge of the administrative scheme within the PSNI.
The MPs launched their inquiry after it emerged that on-the-runs had been sent letters from the NIO informing them that they were not wanted.
On Thursday, Justice Minister David Ford reiterated that his Department had no part in the scheme.
"My Department has had no involvement with the scheme and so does not have information on the requests made nor the content of and names of those to whom letters issued," he said.
"That information is held by the NIO and the Secretary of State made a statement to the House of Commons last week on the number of cases considered by the NIO."
He added: "The Secretary of State made the welcome announcement on 7 March that the scheme had ended but I am conscious that there are many issues to be resolved and I am as keen as anyone to ensure that is the case.
"In doing so, my main concern is to look after the interests of victims and their right to seek truth and justice."