Stones pulled his hamstring during City's 2-0 win over Leicester City at the King Power Stadium on 18 November, with the injury likely to sideline the centre-back for up to six weeks.
The Daily Mirror says City boss Pep Guardiola will prioritise bringing in defensive reinforcements in the January transfer window, with Vincent Kompany, Nicolas Otamendi and the out-of-favour Eliaquim Mangala the only fit centre-backs at the Etihad Stadium.
Guardiola said after the win over Leicester that he had a "real problem" in the centre-back position heading into a hectic schedule of fixtures.
"It is unfortunate for us and for John and I think he might be out for six weeks with that type of injury," the Spaniard told City's official website.
"We have to suffer as he plays for 90 minutes, and 90 minutes and 90 minutes. We don't have a lot of central defenders. But the spirit in the team and with all the players is good so we are going to find a solution."
Liverpool have been linked with a £60m ($79m) move for Van Dijk in January, but the Mirror says City could table a rival bid for the Netherlands international.
Guardiola added: "We're going to discuss when January is coming. We are going to think about it, but not because John is in that position.
"To play in all the competitions, you need a bigger squad because the players don't have time for regeneration before the next game."
Liverpool publicly ended their interest in Van Dijk in June after Southampton threatened to report the Reds to the Premier League over an alleged illegal approach for the player.
Reds owners Fenway Sports Group assured Southampton that they would not make any further approach for the player unless they were invited by the south coast club.
However, recent reports have suggested that the Merseyside club will launch a fresh attempt to sign Van Dijk in January.