Manchester United also asked about the midfielder earlier this month and remain interested, sources said, but have not yet been able to reach an agreement with Atletico over a fee.
Saul, 26, has started both of Atletico's league games so far this season but has lost his status as a key player for coach Diego Simeone over the last year.
The LaLiga club are not planning to move for a midfielder to replace him, and their final signing of the transfer window is expected to be the imminent arrival of Brazil forward Matheus Cunha from Hertha Berlin.
Sources have told ESPN that Chelsea coach Thomas Tuchel isn't pushing hard for the move as he is happy with his squad. However, sources added that he is aware the situation could change next season and this is why the club are keen on a loan deal this summer.
Saul is an Atletico academy product who became a first team regular in the 2015-16 season.
His knack of scoring spectacular goals in high-profile games -- such as a bicycle kick against Real Madrid in February 2015 and a solo strike in a 1-0 win over Bayern Munich in the 2016 Champions League semifinals -- helped raise his profile.
Simeone came to value his athleticism, work rate and versatility, and the club tied him down to a nine-year contract in 2017.
He started just 22 LaLiga games, often featuring on the left-hand side, and sources told ESPN that his performance over the last 18 months had not met expectations.
Saul has been frequently linked with a move to the Premier League over the years, with a belief his box-to-box skillset would be well suited to English football.
Information from ESPN's James Olley was also included in this report.