Despite spending over £200m on players so far this summer, Pep Guardiola is determined to strengthen his defensive options and sees Evans as the perfect player to provide cover for Vincent Kompany, John Stones and Nicolas Otamendi.
Last week, the former Manchester United defender emerged as a surprise target for City, who had a first offer in the region of £18m knocked back and were reportedly ready to up their bid to around £22m.
However, with Albion manager Tony Pulis keen to retain the services of the 29-year-old and with less than 10 days left in the transfer window, the former Barcelona and Bayern Munich manager is ready to look elsewhere.
According to the Independent, Gibson has emerged as City's preferred alternative to Evans mainly because, as is the case with the Northern Ireland international, he would allow Guardiola to add a home-grown or 'locally-trained' player to comply with Premier League and Uefa regulations on squad compositions.
A product of Middlesbrough's academy, Gibson represented England at every junior level from U17 up to U21 before being into the England squad by Gareth Southgate in March.
City are believed to be considering a bid in the region of £20m for the 24-year-old, who started every game of Middlesbrough's Premier League campaign last term and who was in the starting XI in all four of Boro's league games so far this season.
Gibson has attracted interest from a number of Premier League clubs, including Chelsea, but Boro are expected to fight to keep him at the Riverside Stadium as they look to bounce back to the top-flight at the first time of asking.
City were reportedly ready to almost double Evans' weekly wages from £75,000 to £140,000 in a bid to lure him back to Manchester, where he won three Premier League titles in nine years at United, but fear they might have to spend around £30m for the defender.
"We don't want to sell him, we don't need to sell him," Pulis said before last Saturday's (19 August) trip to Burnley. "We're not desperate to get the money in, so it'd have to be a very, very good offer. And then we would sit down and talk to Jonny about the situation."