Theresa May said the move would be "appropriate" after a watchdog said there were questions over "honesty and integrity".
Mr Mitchell lost his job following allegations that he called police officers "plebs" when they would not let him cycle his bike through Downing Street's main gate.
Mr Mitchell has always protested his innocence, although he has admitted swearing at police during the altercation.
The latest development centres on a meeting between three Police Federation officers and Mr Mitchell at the height of the row.
Immediately after their discussion, one of the officers called for Mr Mitchell to resign, saying he had refused to elaborate on what happened, Unknown to him Mr Mitchell had recorded their conversation. The tape shows he clearly denied saying the word "pleb".
An investigation by West Mercia Police concluded that there was "no deliberate intention to lie" from the officers. But in a statement, the deputy chair of the Independent Police Complaints Commission Deborah Glass disagreed.
"The evidence is such that a panel should determine whether the three officers gave a false account of the meeting in a deliberate attempt to support their … colleague and discredit Mr Mitchell, in pursuit of a wider agenda," she said.
"In my opinion, the evidence indicates an issue of honesty and integrity, not merely naïve or poor professional judgment."
The report made for "troubling reading", Mrs May told MPs on the Home Affairs Committee yesterday. Asked if the chief constable of West Mercia Police should apologise to Mr Mitchell, Mrs May said: "I think that would be appropriate."
She added: "The police need the trust of the public. These sorts of incident will strike at the heart of that issue of trust."
The chair of the committee Labour MP Keith Vaz said afterwards that West Mercia chief constable David Shaw would be hauled in front of MPs next week.
In a statement, Mr Mitchell said that he and his family had "waited in vain" for the officers to be held to account.