The Barcelona forward almost joined former boss Pep Guardiola at the Etihad in August before opting to stay at the Nou Camp but it is looking increasingly likely he will leave at the end of the season.
According to The Sun, City offered the 33-year-old a five-year deal worth £600m last summer but have now presented the Barca captain with a new package of £430m.
The report claims the Premier League side are not prepared to match their previous offer due to the financial climate and that Messi is another year into his career.
The forward is out of contract at the end of the 2020-2021 campaign, meaning City would not have to pay a transfer fee.
It is also reported that City want Messi on a two-year deal at the Etihad with a 12-month option before another two seasons in the MLS at sister club New York City.
The potential package could see him commit to eight years with the forward also carrying out ambassadorial duties with the clubs owned by the City Football Group.
Messi is Barca’s highest earner on £500,000 a week (£26m a year) but he and his team-mates to wage reductions in November to help ease the financial burden of the pandemic.
Paris Saint-Germain are also said to be interested in the six-time Ballon d’Or winner.
The French side beat Barca 4-1 in the first leg of their last 16 Champions League tie on Tuesday with Kylian Mbappe scoring a hat-trick.
Messi looked frustrated as side suffered another damaging European defeat after the Catalans lost 8-1 against Bayern Munich in the quarter-finals last season.