The 24-year-old defender submitted a transfer request after Manchester City expressed interest, but they decided against it since the price was too high.
Although sources at Chelsea are optimistic that the trade will be completed, a statement from Brighton said that “no agreement has been reached with any club.”
Cucurella joined Brighton from Getafe and stayed there for one season.
After defender Kalidou Koulibaly and striker Raheem Sterling, he would be Chelsea’s third summer acquisition.
The Blues and Aston Villa have also agreed on a £20 million agreement for Carney Chukwuemeka, a teenage England junior international.
When champions City made a £30 million approach for Cucurella last month, the Seagulls rejected it. They set a standard of £50 million; the price Arsenal paid for England defender Ben White last summer.
According to club sources, a more substantial offer would have to be made for the club to contemplate selling Cucurella.
But if he were to leave the Amex this summer, he would join midfielder Yves Bissouma in leaving for Tottenham in a £25 million deal with add-ons.
Cucurella, a product of the Barcelona academy, has one senior cap for Spain and was a member of the under-23 squad that won silver at the 2020 Tokyo Olympic Games.