The Blues have already had made an offer believed to be worth around £27 million for the unsettled striker, but were knocked back last month.
Things even threatened to turn ugly as reports claimed Chelsea had offered either David Luiz or Juan Mata as part of the deal, something the west London club were forced to deny, claiming the story had come from the Manchester United camp.
Chelsea boss Jose Mourinho had hinted he wouldn't raise his offer for Rooney, whose future at Old Trafford looks uncertain after he apparently handed in the second transfer request of his Man Utd career towards the tailed of last season.
United manager David Moyes also insisted that Rooney wouldn't be sold but now Chelsea look set to test the Scot's resolve with a vastly improved second offer.
Metro claim that Chelsea will offer £40 million to land Rooney, who has two years left on his contract at Old Trafford.
The paper also claims that should Rooney, who is currently nursing a hamstring injury, arrive at Stamford Bridge it is likely that one of either Fernando Torres or Demba Ba could leave the club.
Previously Mourinho has in sited he is "calm" over the prospect of landing Rooney this summer, and said that if the deal failed to materialise he was "happy" with what he had.