Tony Blair advised Rebekah Brooks to launch a "Hutton style" inquiry into phone hacking at the News of the World at the height of the scandal over the issue, according to an email that has emerged at the Old Bailey trial.

The revelation emerged in an email that was read to the jury in the hacking trial on Wednesday, and followed what Brooks said was an hour-long phone call.

According to the email, sent the day after the News of the World's final issue and six days before Brooks was arrested, Blair also told her he was "available" to her and Rupert and James Murdoch as an "unofficial adviser" on a "between us" basis.

The advice was said to have been given on 11 July 2011 and contained in an email she sent at 4.20pm to James Murdoch, the then executive chairman of News International.

According to Brooks's note, Blair advised her to set up an "independent" inquiry, suggesting it could have "outside counsel, Ken Macdonald [the former director of public prosecutions], a great and good type".

He said the inquiry would be "Hutton style" – a reference to Lord Hutton's inquiry into the death of David Kelly – and would "clear" her, but warned that "shortcomings" would have to be accepted as a result of the report.

Rebekah Brooks and Tony Blair in 2004. Photograph: Fiona Hanson/PA

According to the email the advice was given in an hour-long phone conversation. Blair advised her to "tough up" and not to make any "rash short-term solutions as they only give you long-term headaches." He also told her to "keep strong" and advised her to take "sleeping pills".

Prosecutor Andrew Edis read out the entire email exchange between Brooks and James Murdoch to the jury as part of the formal conclusion of the Crown's case.

After finishing in the email he turned to the jury to simply say "Well, that's that" before moving on to the next piece of evidence.

Brooks told James Murdoch in the email: "I had an hour on the phone to Tony Blair" and then proceeded to outline the points he had allegedly made in the conversation.

"1. Form an independent unit that has an outside junior counsel, Ken Macdonald, a great and good type, a serious forensic criminal barrister, internal counsel, proper fact checkers etc in it. Get them to investigate me and others and publish a Hutton style report," she said.

"2. Publish part one of the report at same time as the police closes its inquiry and clear you and accept short comings and new solutions and process and part two when any trials are over.

"3. Keep strong and definitely sleeping pills. Need to have clear heads and remember no rash short term solutions as they only give you long term headaches.

"4. It will pass. Tough up.

"5. He is available for you, KRM [Rupert Murdoch] and me as an unofficial adviser but needs to be between us," she wrote.