The 17-year-old left-sided player was due to undergo a medical at Stamford Bridge on Tuesday before agreeing personal terms on the switch.
Chelsea manager Jose Mourinho is impressed by Elmer's potential after his emergence in Switzerland.
And he is keen to add youth to Chelsea's squad as well bring in expensive new signings.
Elmer, who will join the Blues on a three-year deal, is a highly-rated Switzerland under-17 international and was spotted by Chelsea scout Gwyn Williams during a scouting trip to watch a Spanish player.
He will join the Londoners as part of the reserve team set-up and continue his education while with Chelsea.
Although the Premiership champions have the funds of owner Roman Abramovich at their disposal, Mourinho has told the club he also wants to bring the world's best young talent to Stamford Bridge and Elmer represents the first move in this direction.
That policy will be strengthened yet further when Chelsea finally agree a compensation package for Tottenham's suspended sporting director Frank Arnesen.
Talks between to the two clubs are still continuing and may now revolve around the sale of reserve team striker Carlton Cole to Spurs as part of the settlement.
Switzerland is fast becoming a top hunting ground for English clubs with Arnesen having brought Reto Ziegler to Spurs last year and Arsenal ready to give Philippe Senderos a new deal at Highbury.