The Brazilian flew to London on Tuesday afternoon to complete a medical ahead of his long-awaited move from Roma.
The two clubs are understood to have agreed an initial fee of £17.5m with the Italian side, with another £5.2m in add-ons.
Photos emerged on Twitter on Tuesday night of Palmieri holding up a Chelsea shirt in what appeared to be his presentation as a Chelsea player.
Palmieri will provide much needed competition at left-back, with Marcos Alonso having previously been Antonio Conte's only senior, experienced option.
Official confirmation of the deal from Chelsea is expected to follow soon.
30 January 2018