The Citizens are close to securing a £50 million move for the Juventus centre-half.
Juventus' Leonardo Bonucci could be on his way to Manchester City
The newspaper claims City have upped the ante over the last 24 hours, and they are set to meet the Old Lady's valuation of their star centre-back, a fee the Mirror claim to be £50 million.
Should City fork out that sum, it would equal the record transfer fee for a defender, an accolade currently held by David Luiz of PSG.
Bonucci battling it out with Robert Lewandowski in the Champions League
Guardiola is reportedly keen to make Bonucci the bedrock of his defence as injuries continue to plague club captain Vincent Kompany.
During his unveiling press conference, the former Barcelona and Bayern Munich coach urged the Belgian to focus on his fitness if he wants to have a future at the Etihad Stadium.
Manchester City's Vincent Kompany waves to fans as he walks off to be substituted after sustaining another injury
"What do I think about Kompany? People in England knows his quality, maybe better than me. My dream for Vincent Kompany is to be fit,' Guardiola said as quoted by the Daily Mail.
"We cannot ask about Kompany, in the last two years he's played not too many games. To be fit [is his goal]."
Chelsea manager Antonio Conte was also said to be very keen to land the Italian international, having coached the player at Juventus and also with Italy as recently as last month at Euro 2016.
However, City's offer of £140,000 per week could well be the deal clincher for the Italian.
Bonucci is expected to cost £50 million
Bonucci began his career at Inter Milan, but he only made one appearance before establishing himself at Bari, where he was picked up by Juventus in 2010.
He has since become one of the world's most highly rated centre-halves, making 198 league appearances for the Old Lady, and 62 for his country.