Pope, the veteran Vale striker, claimed in a mocking tweet last summer that he would score 40 a season if he played against City defender Stones every week.
He came good as he cancelled out an Oleksandr Zinchenko strike at the Etihad Stadium but the League Two minnows were ultimately outclassed as Sergio Aguero, Taylor Harwood-Bellis and Phil Foden netted for the Premier League champions.
Two of City’s goals, for Aguero and Harwood-Bellis, required lengthy VAR checks before being allowed but neither were controversial.
Stones thought he had even levelled up his personal battle with Pope by grabbing City’s third but it was eventually awarded to Harwood-Bellis.
Stones also returned after a month out with a hamstring injury and all eyes were immediately on him and Pope, who created an early chance for Scott Burgess only for the shot to be deflected wide.
That minor scare over, City settled into their usual rhythm and offered Vale little of the ball. Foden had a shot blocked and then headed straight at Scott Brown.
It came as no surprise when City took the lead after 20 minutes, although Zinchenko’s long-range shot took a big deflection off Leon Legge.
David Silva should have doubled City’s lead but remarkably hit the bar from point-blank range and Aguero failed to tap in a simple chance on the rebound.
Zinchenko then had another effort diverted wide before Vale, out of the blue, grabbed a shock 35th-minute equaliser.
Once again City’s vulnerability to the counter-attack was highlighted as former Liverpool trainee David Amoo broke clear down the right.
Amoo then picked out Pope with a precise cross and, coping well in the spotlight, the 34-year-old stole ahead of Harwood-Bellis to plant a firm header past Claudio Bravo.
Inevitably observers had to check Stones’ positioning, but the England centre-back was not at fault.
City responded in clinical fashion as Ilkay Gundogan chipped a pass out wide to Foden and the youngster’s volleyed cross was turned in by Aguero.
With a VAR check pending for possible offside against Foden celebrations were muted, but the goal stood.
City looked to ramp things up after the break and Aguero fired over before Brown saved from a Bernardo Silva header and two strikes by Foden.The third goal came just before the hour and it seemed Stones had struck back against Pope by tucking away from five yards. However, on close inspection, amid a long VAR check for possible offside, it transpired the ball had gone in off Harwood-Bellis.
City then had the game under control but Stones’ battle with Pope was not over.
Pope managed to get a touch to an ambitious pass across his own box by Stones and diverted it onto the path of Tom Conlon but the Vale substitute headed wide.
Pope was substituted soon after but City wrapped up the game when Angelino cross for Foden to add the fourth in the 76th minutes.