Nottingham Forest are keen to strengthen before Tuesday's deadline and are preparing to offer £40m plus a potential sell-on clause to clinch a deal for Armstrong, as first reported by The Athletic.
It would be a major surprise should Everton sanction a deal, but there is hope at the City Ground that Oliver Glasner's side can pull off the move to prise away the elegant England Under-19 international, who is regarded as having a big future ahead of him.
Armstrong, 19, has made 23 appearances for Everton and started in their 2-0 Premier League win over Crystal Palace last week, providing the assist for Kiernan Dewsbury-Hall's opener. He was also named in the starting XI for Saturday's trip to Bournemouth. As yet, there are no indications as to whether Everton will agree to sell, with manager David Moyes having spoken carefully about not placing too much pressure on the teenager following his emergence this season.
The move would follow a familiar path for Forest, who paid Newcastle United £116m for Elliot Anderson earlier in the summer after the midfielder made a similarly rapid rise to first-team prominence. Glasner is understood to see parallels between the two situations, with Armstrong seen as a player capable of following Anderson's trajectory.
Central midfield remains an area of relative strength for Everton following the arrivals of Hayden Hackney, Merlin Röhl and Christian Nørgaard this summer, while James Garner and Tim Iroegbunam are also closing in on returns from injury, which may make Moyes more open to negotiating than would otherwise be the case.
The interest in Armstrong comes as Everton also brace for a fight to keep hold of Iliman Ndiaye, who has attracted interest from Manchester City in the closing days of the window. Forest's pursuit also follows two failed attempts to sign Tottenham Hotspur midfielders Lucas Bergvall and Pape Matar Sarr earlier in the window, with Spurs rejecting their bid for the Sweden international and Sarr instead weighing up options that include Galatasaray, Beşiktaş, Crystal Palace and Brentford.
Having drawn a blank on both targets, Forest have now switched their full attention to Armstrong, and are understood to be making progress in talks with Everton over a structure that could yet see a deal completed before deadline day.