The Mirror report that the 23-year-old midfielder has attracted interest from Stoke, Wigan and Bolton but the Scottish champions now look ready to put in a first offer.
Contract negotiations between Johnson and Leeds have broken down and he could be available for less than £500,000.
With his current deal expiring at the end of the season, it means the Championship side could decide to cash in rather than lose him for free.
Johnson is understood to be enthusiastic on a move to the Premier League but would consider heading to Scotland if it brought him the chance of Champions League football.