It would be a major coup for R's boss Mark Hughes if he were to land Johnson, as there a considerable number of clubs who are after his services.
Cross London rivals West Ham also said to be linked along with fellow newly promoted side Reading and clubs in the UAE have also shown an interest.
The 31-year-old will be on the move from Fulham and Hughes is keen to add his quality and experience to his squad, as he tries to ensure the club do not find themselves in a relegation fight again at the end of next season.
Johnson is not in a rush though and the R's will have to put a package together which at least matches what others have to offer if they are to land him.