Blair also offered to be an "unofficial" advisor to Murdoch and his son James at the height of the scandal which led to the closure of the tabloid in July 2011.
An email written by Rebekah Brooks, then chief executive of Murdoch's British newspaper group, News International, described an hour-long phone call with Blair in which he allegedly told her to "tough up".
The email was shown to the jury at the end of almost four months of prosecution arguments in the phone-hacking trial, in which Brooks is expected to take the stand later this week.
The 45-year-old denies charges of conspiring to illegally access voicemails, bribing public officials and trying to hide evidence from police.
Brooks' email was addressed to James Murdoch, Rupert Murdoch's son and then chairman of News International, and was dated July 11, 2011, the day after the scandal-hit News of the World published its final edition.
Writing in note form, Brooks described how Blair allegedly told her: "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."
He also reassured her that "it will pass. Tough up".
On July 15, Brooks resigned and on July 17, she was arrested for the first time on suspicion of phone hacking.
Brooks wrote that Blair said he was "available to you (James Murdoch), KRM (Rupert Murdoch) and me as an unofficial adviser but needs to be between us. He is sending more notes later".
The email also says Blair recommended that Brooks set up an independent inquiry to investigate senior News of the World managers.
Brooks wrote that Blair told her to "publish part one of the report at same time as the police closes its inquiry and clear you and accept shortcomings and new solutions and process and part two when any trials are over"
- Brooks prepares to testify -
Brooks is expected to give evidence on Thursday when lawyers for the seven defendants begin making their case. They were due to begin on Wednesday but were delayed by legal arguments.
Brooks denies conspiring to hack phones while she edited the News of the World between 2000 and 2003, specifically conspiring to illegally access the voicemails of missing schoolgirl Milly Dowler, who was later found murdered.
The allegation in 2011 that Dowler was targeted proved the final straw in a slow drip of revelations about phone hacking at the News of the World, prompting Murdoch to summarily shut down the 168-year-old Sunday tabloid.
Brooks, once one of the most powerful women in British media and so close to Murdoch that commentators dubbed her his "fifth daughter", is among three defendants facing charges of phone hacking.
The others include Andy Coulson, her former lover and deputy at the News of the World, who replaced her as editor in 2003, and former managing editor Stuart Kuttner. They all deny the charges.
Prime Minister David Cameron will be watching closely when Brooks and Coulson take the stand, given his once close ties to both journalists -- and the huge potential embarrassment.
Cameron frequently socialised with Brooks and her husband and hired Coulson as his communications chief after he left the News of the World in 2007.
In related charges, Brooks is accused of hiding evidence from the police, allegedly aided by her husband, former PA Cheryl Carter and head of security Mark Hanna.
She faces a further charge of conspiring to pay a defence ministry official for stories.
Coulson is also accused of bribery along with Clive Goodman, the News of the World's former royal editor who was jailed for hacking in 2007. They both deny the charges.
