The Tottenham Hotspur defender has been linked with a potential move to the Premier League champions over the past week as Antonio Conte looks to improve his wing-back options.
Manchester United boss Jose Mourinho is also thought to be in the market to sign a left-back after Luke Shaw struggled to impress his new manager in the 2016-17 season.
Rose has established himself as a key member of the Tottenham team since Mauricio Pochettino took over the reins at the north London club from Tim Sherwood back in 2014.
The England star missed the second half of the 2016-17 season due to injury as Spurs finished in second place in the table behind eventual champions Chelsea.
Sky Sports News reporter Solhekol has confirmed that Rose is keen on a move to Chelsea or Manchester United before the close of the summer transfer window.
“This is a story that’s got everyone talking. Danny Rose wants to join either Chelsea or Manchester United before the transfer deadline, if Spurs are prepared to let him go,” Solhekol told Sky Sports.
“The England left-back says time is running out for him to win trophies and he wants to see more top class players at Spurs.”
Spurs have already sold right-back Kyle Walker to Manchester City in a world record £52m fee for a defender last month.
Tottenham finished seven points behind Conte’s Chelsea team in the title race last season.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news