Chelsea scouts have been sent out to Germany to watch the 20-year-old Turkish international on numerous occasions this campaign, while Liverpool and Arsenal have also been monitoring the player.
Chelsea value the player at a little under £8 million, but the Bundesliga club are demanding closer to £12 million for their prize asset.
Hamburg finished seventh last season but have struggled this year, falling to within a point of an automatic relegation spot.
Given his rising stock and Hamburg's current lack of competitiveness, Calhanoglu is looking to force through a move in the summer and has assisted the help of Muzzi Occan, an agent based in the United Kingdom, to expedite a switch to the Premier League.
With the likes of Eden Hazard, Oscar, Marco van Ginkel, Willian on the books at Chelsea, the London club have supreme depth at the attacking midfield position. For that reason, signing Calhanoglu would be a strong indication of how highly they rate the Turk.