The People reports that the signing of the Santos striker will be the first topic of conversation when Mancini meets with new director of football Txiki Begiristain.
The paper notes: "Begiristain spent months trying to sign Neymar when he was sporting director at Barcelona between 2008 and 2010 and believes the dialogue they opened then gives him a head start over rival clubs."
Barcelona remain favourites to sign the 20-year-old wonderkid as they have already put down £6m for first refusal but Santos insist that no deal has been done and Mancini remains hopeful.
"I don’t know if it is true that Barcelona have already made a down-payment for Neymar," the paper quotes Mancini as saying.
"I think when Txiki starts to work here next week we can start to talk about the future for us because this is very important.
"Neymar is a Santos player and I don’t know what can happen with him in the future.
"But I am very happy to work now with the new sporting director because he gives us the chance to move very quickly regarding some players."
On Saturday, Neymar scored a hat-trick to lead Santos to a 4-0 win at Cruzeiro in the Brazilian championship, where his performance earned him a standing ovation from the opposition supporters.
However, Mancini's own future remains in doubt as City's owners are believed to be unhappy with the club's performance in the Champions League and also with reports that Mancini held secret talks with Monaco last year.
Many of Sunday's papers report that Pep Guardiola has instructed 'his people' that he is ready to return to work next season after taking a year out of the game.
The former Barcelona boss could be a perfect fit for City given his past relationship with Begiristain.
Much of City's success has come from paying huge wages and transfer fees but Premier League owners have been holding talks to try and limit the amount the players and their agents get so they can pocket more of the league's latest £5 billion TV deal for themselves.
The Daily Mail reports that a formal outline could be agreed later this month when the Premier League chairmen meet on November 15.
The paper reports that: "Among the more radical proposals is that member clubs should be made to break even, something that even UEFA’s fairplay rules do not insist on. And West Ham chairman David Gold wants a points deduction for those whose debt exceeds a certain level of turnover.
"Significantly, there is a will among even the richest owners — (Roman) Abramovich at Chelsea and Sheik Mansour at Manchester City — to stop making huge losses. And the new breed of American owners — John W Henry (Liverpool), Ellis Short (Sunderland), Stan Kroenke (Arsenal), the Glazers (Manchester United) and Randy Lerner (Aston Villa) — are desperate to see a return on their investment."
To other rumours and Theo Walcott is heading towards the exit at Arsenal because of Arsene Wenger’s continued refusal to start him in Premier League games, reports the Times.
The paper says: "Liverpool are considering an £8m bid for him in January, when Walcott, who also interests Paris Saint-Germain, can sign a pre-contract agreement with a foreign club."
Liverpool are also ready to launch a bid for Klaas-Jan Huntelaar after discovering that the Dutch striker may be available for just £6 million. Huntelaar is out of contract at Schalke 04 in the summer and is expected to reject a final offer from the Germans (Daily Mirror).
Emmanuel Adebayor is wanted by Atletico Madrid as he looks for a way out of Tottenham – just 10 weeks after sealing a move to White Hart Lane. Atletico, who are enjoying an outstanding start to the La Liga season, are weighing up a bid for Adebayor as they seek potential replacements for highly-rated Radamel Falcao who is wants by many of Europe's top club. (The People)
Manchester United and Arsenal face a struggle to sign German star Lars Bender. Both clubs are planning moves for the £20million-rated Bayer Leverkusen midfielder but the club says they will not sell. (The Sun)
Turkish giants Galatasaray want to sign Liverpool's Joe Cole in loan in January - and are willing to pay the majority of his reported £100,000 a week wages, (The People)
Finally, Blackpool have lined up Paul Ince to be their new boss. Ince's son Thomas is the club's star player. (The People)