Archer was released by Spurs a year ago but has since established himself as Millwall's number 1, ousting Republic of Ireland international David Forde.
The 23-year-old, who is coached by former Premier League goalkeeper Kevin Pressman at The Den, has kept 14 clean sheets in 33 League One appearances this season - helping the Lions to fifth in the table with five games still to play.
But with his contract set to expire in a year's time Archer could prove a cost-effective replacement for Everton's Tim Howard, who has agreed a return to his homeland with Colorado Rapids.
Howard lost his place to Joel following a questionable run of form earlier in the campaign.
Should Everton move for Archer, who was born in London but has caps for Scotland Under-21s, he would be the latest in a long line of Football League prospects to be plucked by the Toffees.
Brendan Galloway (MK Dons), Mason Holgate (Barnsley) and Matthew Foulds (Bury) have all moved to Goodison Park in recent seasons, though England centre-back John Stones (also Barnsley) has proved the club's most successful lower league recruit so far.