The Seagulls rejected two bids from Premier League rivals Arsenal for the Ecuador international, while Chelsea were also reportedly interested.
Caicedo last week informed Albion he wished to leave in an open letter posted on social media, prompting the club to give him time off until after the transfer widow closed on Tuesday evening.
The 21-year-old said on Instagram that he was "proud to be able to bring in a record transfer fee for Brighton" but did not specify which team he wanted to join.
He hoped Albion would understood why he wished to "take up this magnificent opportunity".
The Seagulls were adamant they had no intention of selling and, after successfully fending off interest for the time being at least, have now welcomed back Caicedo to the club's Elite Football Performance Centre in Lancing.
Caicedo was absent for Sunday's 2-1 FA Cup victory over Liverpool at the Amex Stadium.
Head coach Roberto De Zerbi said before and after the fourth-round tie that he wanted the player to finish the season with the Sussex club.
Caicedo, who arrived from Ecuador's Independiente del Valle in 2021 and is under contract until 2025, has made 26 league appearances for the high-flying Seagulls, scoring twice.
Albion return to action when they host Bournemouth in the top flight on Saturday.
De Zerbi's side sit sixth in the table, five points behind fifth-placed Tottenham but with two games in hand.