The two clubs are close to an agreement over a final fee and the 23-year-old is expected to make a switch to Pep Guardiola's team.
City are keen to get the deal over the line before they fly out to the Club World Cup on June 12 and negotiations with Wolves are now accelerating, but not yet complete.
Once finalised, Ait-Nouri is expected to leave the Algerian national team camp to complete his medical.
Angers still hold a 50 per cent sell on clause, that Wolves can buy out for around £9.5million.
That has yet to be bought out, but it is understood Wolves will do that before completing Ait-Nouri's sale to City.