Blues had accepted a £33m offer from Atlético Madrid last week, but there were significant concerns about whether the England international was willing to make the move.
However, on Sunday night, Gallagher, 24, informed Atlético officials that he is happy to make the switch.
The England international is now set to fly to complete his medical and contract signing, ending his 18-year tenure at Stamford Bridge.
Chelsea had proposed a two-year contract with an option for an additional year, but Gallagher found it unfavorable due to both the contract length and the lesser role he was offered.
A natural divide had emerged between Gallagher and Chelsea, as he was seen as a poor fit for new manager Enzo Maresca’s possession-based style of play and would not be a regular starter under the Italian.
Gallagher, who played five matches at Euro 2024, has less than one year remaining on his current Chelsea deal.
Chelsea preferred to sell Gallagher overseas rather than to an English rival, which allowed Atlético to secure a cut-price deal.