The Blues identified the 23-year-old as their top target late last year and although they have considered several other left-back options including Ajax's Nicolas Tagliafico and Real Madrid's Sergio Reguilon, they have remained hopeful throughout that an agreement for Chilwell could be reached.
Negotiations are ongoing but sources close to the transaction suggest a fee in the region of £50m is now being discussed with Leicester willing to do business.
They had adopted a strong negotiating position based on both manager Brendan Rodgers' desire to keep the England international and the fact Chilwell has a contract at the King Power Stadium until 2024.
However, Chilwell is keen to move to west London and while he has not tried to force through a transfer, there has been a growing expectation in recent days that Leicester would soften their position on allowing him to leave.
Like many Premier League clubs, Leicester have been hit by the financial impact of COVID-19, a blow compounded by missing out on Champions League qualification on the final day of the season as, ironically enough, Chelsea snatched fourth place.
Chilwell could miss the start of next season due to a heel problem but it is not expected to hinder any medical. Personal terms have not been finalised but they are not expected to be an issue.
Chilwell is likely to be joined by Kai Havertz as the Blues close in on a deal with Bayer Leverkusen worth in excess of £75m, with the pair expected to join fellow new signings Hakim Ziyech and Timo Werner in a devastating spending spree which head coach Frank Lampard hopes will turn the club into title contenders.