Last Friday, the left-sided centre-back made a surprise appearance off the substitutes' bench as the Premier League champions started their top-flight campaign with a 3-0 victory.
That outing came amid speculation that City are open to offers for the Spain international this summer, a consequence of having already signed Josko Gvardiol.
As it stands, though, realistic offers from European-based clubs have been in short supply, handing an opportunity to teams from the Saudi Pro League.
Earlier this week, it was claimed that City had opted to accept an offer from Al-Nassr, who are eager to match the signings made by other clubs in the division.
Although Alex Telles, Marcelo Brozovic, Seko Fofana and Sadio Mane have all joined Cristiano Ronaldo at Al-Nassr, they lost to Al Ettifaq in their opening league game.
However, according to The Sun, Real are toying with the idea of making their own bid for Laporte over the coming days.
At one stage earlier this summer, Carlo Ancelotti had seemingly been in a position where his squad for the 2023-24 campaign was effectively complete.
Serious knee injuries to Thibaut Courtois and Eder Militao have changed that outlook, though, with Chelsea goalkeeper Kepa Arrizabalaga having already been acquired to replace the former.
Real are down to three options for the middle of the backline and they are allegedly ready to explore the possibility of signing Laporte.
That said, it is unclear whether Los Blancos would first have to part ways with a current member of their squad courtesy of Man City being unwilling to consider a loan deal like Chelsea were with Arrizabalaga.
Laporte has less than two years remaining on his contract, yet City will still expect to recoup in the region of £30m for the 29-year-old.
Al-Nassr are also said to have been in the market for Manchester United's Eric Bailly, who remains an alternative option should Real make the breakthrough for Laporte.