The Tractor Boys renewed their interest in 26-year-old midfielder Norris as soon as the transfer window reopened.
However, the Pilgrims have reportedly valued their prize asset at £2million, due to the fact half of any transfer fee is likely to go to Bolton under the terms of Norris' transfer five years ago.
Town boss Jim Magilton is likely to make an improved offer for the player although Ian Holloway's Leicester are also tracking the situation.
Argyle are also facing the prospect of losing versatile England Under-19 midfielder Dan Gosling to Everton. The Toffees have been linked with the highly-rated 17-year-old for several months and are said to be lining up a January swoop. Gosling was not involved in the matchday 16 as Argyle lost 1-0 at Cardiff in the Championship on New Year's Day, fuelling speculation about his possible departure.
Doubt has also been cast over Manchester United winger Lee Martin's loan deal at Home Park. The 20-year-old has impressed with two goals in 12 league outings for the Greens, but could now be lured to Leicester or Wolves instead.
Other transfer business is close to being completed. Hungary midfielder Akos Buzsaky's loan deal at QPR is set to be made permanent for £500,000, while veteran striker Barry Hayles is costing Leicester £150,000, and some of that money will be used by Paul Sturrock to complete a permanent £210,000 deal for Wales striker Jermaine Easter, currently on loan from Wycombe.
Meanwhile, Argyle have reportedly tabled a bid for Exeter striker Jamie Mackie, who has been in fine scoring form in the Blue Square Premier this season. The 22-year-old has chipped in with 11 league goals for the eighth-placed Grecians.