According to Paul Joyce of The Times, the Premier League champions are ready to press ahead with their interest in the England international and have lodged their opening offer.
"The offer is below Everton's valuation which stands at £40 million, but Chelsea are ready to continue negotiations as they seek to outflank the England international's long-term suitors Tottenham Hotspur," Joyce noted.
It's suggested initial talks were held between the two clubs on Sunday when they met in league action.
Barkley's future has been the subject of much debate throughout the summer transfer window, with the London giants linked to him. The 23-year-old has less than one year left on his contract at Goodison Park and despite a current injury layoff, he looks set to leave Merseyside before the deadline at 11 p.m. BST on Thursday.
While Barkley has struggled to show his best on a regular basis in recent seasons, he has been improving his creative output, as noted by Squawka Football:
The Everton man is comfortable operating in any position behind a centre-forward. He has the attributes of a central midfielder rather than a wide player or a natural No. 10. Barkley is a strong runner, has a decent engine and is smart in the way he uses the ball.
At his best, he would be a major asset to Chelsea and would potentially benefit from working under Antonio Conte.
It will be fascinating to see whether the Blues are ready to match the valuation placed on the player by Everton, which is substantial for someone who is injured and has such a short amount of time left on their contract.