Armstrong has started every Premier League match for Everton this season and has become an increasingly important part of David Moyes' side.
The 19-year-old's emergence has been particularly notable given that Everton were willing to listen to proposals for him during the summer.
Nottingham Forest made an approach for Armstrong, but the possibility of selling the academy product triggered a strong reaction from Everton supporters. Protests followed, and talks over the proposed transfer subsequently ended.
Dyche, who handed Armstrong his senior debut, believes the interest from Forest was understandable.
“He's a good player; I gave him his debut. I think he's got a really strong chance of becoming a real top player,” Dyche said on The Overlap Fan Debate.
“It's still early, but I can see why clubs were sniffing, and why Forest made the cheeky approach.”
Dyche first brought Armstrong into Everton's senior squad during pre-season in the summer of 2024.
The midfielder made his senior debut later that year in a Premier League away defeat at Tottenham Hotspur, having only just turned 17.
Loan spells at Derby County and Preston North End subsequently helped Armstrong gain regular senior experience and develop his reputation.
He returned to Everton from Preston in January of last season but initially remained on the fringes of Moyes' first-team plans.
That changed this season following injuries to Christian Norgaard, James Garner and Merlin Röhl, giving Armstrong an extended opportunity in the Premier League.
He has taken it, starting every league game and earning praise for his performances.
Dyche also offered a possible explanation for why Everton were prepared to consider Forest's approach, pointing to the financial value of homegrown players under the Premier League's Profit and Sustainability Regulations.
“Maybe, because of PSR and the history of Everton's homegrown talent, they thought it was a bundle of money for PSR and all that,” Dyche said.
The former Everton manager stressed that he was not suggesting the club had definitely intended to sell Armstrong.
“I'm not saying they were never going to get him, by the way, with Forest being the club chasing him, but he's got every chance.”
Dyche was ultimately pleased that the proposed move did not happen.
“I'm pleased he stayed there. He's got every chance of being a top player, certainly,” he said.
Armstrong's development now gives Everton an academy graduate who has moved from being a promising youngster to an established member of the first-team squad.
For a player who made his senior debut at just 17, the rapid progression has already attracted interest from elsewhere.
Dyche believes the teenager has much more to come.