England were without Mark Wood for the fourth day in Manchester after the fast bowler sustained a thigh injury on day three.
Wood, 34, had a scan on Saturday and England are set to provide a squad update on Sunday.
Pope said he “doesn’t know” the extent of the injury, but it seems unlikely Wood will be able to play in the second Test at Lord’s which begins on Thursday.
That would leave Olly Stone in line for his first Test in more than three years, with England possibly looking to add cover in their squad in the shape of Sam Curran, uncapped Essex bowler Sam Cook or Leicestershire 20-year-old Josh Hull.
And Vaughan, who captained England to victory in the 2005 Ashes, said it was right to be cautious with Wood, especially with an eye on the tour to Australia in 2025-26.
“Mark Wood is the key,” said Vaughan. “Ben Stokes is a talisman and an incredible cricketer who England are missing, but they miss Mark Wood more because he can bowl 90mph so consistently.
“I can’t see him playing at Lord’s, maybe not at all this summer, because he needs to be fit for Australia. If they have him fit out there, they can do something special.”
Pope, 26, has served as vice-captain to Stokes. The Surrey man led England for the first time in Manchester despite only taking charge of one previous first-class match. He will remain as captain for the rest of the three-match series.
Stokes, who has a hamstring injury, has been with the England squad throughout the first Test and will be for the final two matches.
“It was different, more so in the field,” said Pope. “There were some good lessons learned for me.
“I think Stokesy was bored at times. He’d much rather be playing. He was great. Every now and again I’d pick his brain, more than he comes to me.
“He wanted to give me my own space to do it my own way, but I know there will be conversations with him and [McCullum] while we’re on the pitch about potential plans for different batters, which is great to have when we come off for a break.”