The developments came ahead of the release of the provincial auditor general's report into government travel slated for Thursday.
In a letter published in the Edmonton Journal and Calgary Herald, Redford acknowledged she made mistakes before quitting as premier last spring to sit as a Progressive Conservative backbencher.
She said her resignation as a member of the legislature for Calgary-Elbow was effective immediately.
"In hindsight, there were many things I would have done differently," she wrote, without being any more specific. "That said, I accept responsibility for all the decisions I have made."
Hancock said there is material in the auditor's review that police should look at.
"The report identifies a number of areas of concern and I think in the interests of completeness and in the interests of the public being fully satisfied that everything appropriate has been done, that it's appropriate to ask that certain issues in the report be investigated," he said in a teleconference with reporters.
Hancock refused to discuss details of the auditor's findings before they are released publicly. But last week, the CBC reported that a leaked copy found that passenger lists on government aircraft were altered so that then-premier Redford could fly alone. She has denied directing her office to make any changes.
Hancock said Redford did the honourable thing in resigning, but, in a statement, he said he was "extremely disappointed" in her actions.
An RCMP spokeswoman said she was limited in what she could say about Hancock's request. Sgt. Josee Valiquette said she didn't believe the Mounties had received an official request yet.
She was unable to say what the procedure would be under these particular circumstances.
Jim McCormick, president of the PC party in Alberta, said it was Redford's "own personal choices that led to her demise."
"She is alleged to have broken government rules, and taxpayer dollars were not treated with the dignity and respect they deserve. She has paid a personal and political price for her mistakes and we appreciate her decision to take responsibility for her actions," he wrote in a statement.
Redford resigned as premier in March after facing increased unrest within her caucus, fuelled by concerns over her leadership style and a $45,000 trip to South Africa for Nelson Mandela's funeral.
She eventually paid back the money for that trip, but only after weeks of pressure and her repeated declarations that she would not.
The criticism over her travel habits persisted.
She admitted flying her daughter, Sarah, and the girl's friends around on a government aircraft.
She recognized those trips didn't follow the rules and repaid about $3,100. She also put a stop to all out-of-province trips on government aircraft until the auditor general had a chance to conduct a review.
The CBC's report last week said the auditor had found "false passengers" were included on flight lists and then dropped at the last minute to keep planes empty. Redford issued a denial via Twitter.
"There was never any directive preventing others from flying on government aircraft when I was a passenger," she said. "In fact, on most occasions that I can recall, when I was on government flights, I travelled with other elected officials, public servants and staff."
In her letter Wednesday, Redford indicated she would offer no further explanations.
"I will leave it to others to analyze and comment on the past. I am sure that I will be asked to weigh in, but I will respectfully decline," she wrote. "It is time to move forward."
Trained as a lawyer, Redford said she will teach and resume work in international development and public policy. She said she will not accept a transition allowance given to members of the legislature who leave their seats.
Calls for an RCMP investigation into Redford's use of government aircraft were almost immediate after the auditor's findings were leaked last week. Opposition Wildrose finance critic, Rob Anderson, said the public has an expectation that politicians who may have broken the law should be investigated.
There were also calls from within her own party for her removal from caucus.
Thomas Lukaszuk, one of the three men vying to become the next leader of Alberta's Conservatives, had called for an emergency meeting to discuss Redford's continued membership in the party.
He said Wednesday that the request for RCMP involvement was the right call.
"Albertans must know that no one is above the law. Whether you are the premier or a minister or an MLA or a public servant — law applies to you just the same," said Lukaszuk. "If there are valid concerns that any laws may have been broken, they need to be investigated just like they would have if you or I had conducted ourselves in the same way."
Fellow leadership candidate Jim Prentice credited Redford for resigning.
"Alison Redford did the right thing and the honourable thing by stepping down and, from my perspective, I certainly wish her and her family all the best going forward," said Prentice, who was campaigning in northern Alberta.
Prentice, the only one in the leadership race who does not hold a seat in the legislature, ruled out seeking election in Calgary-Elbow if he is successful.
The third candidate, Ric McIver, said he realized it was a tough decision for Redford and expects there was probably some pressure from within caucus. "We'll move forward but we will not ignore the lessons of the past," he said.
The man who ran Redford's leadership campaign and was her first chief of staff said he is saddened by how things ended.
"This is kind of the last chapter in something that had a lot of promise that turned out not to be realized," said Stephen Carter, who parted company with Redford following the 2012 provincial election.
He compared her situation to that of another Alberta premier, Don Getty.
"Don Getty was a bad premier, says history, and his oil and gas prices were so low he didn't have a hope in hell," Carter said.
"I think that Alison Redford will be remembered for plane rides and automobile trips and things like that when she should be remembered for someone who started to move us in the right direction financially."
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