According to the BBC and Sky Sports, City will pay an initial £32m for the 22-year-old Brazilian, with a further £2m in potential add-ons.
Allan, who also plays as an attacking midfielder, joined Palmeiras' youth set-up in 2019 and made his senior debut in January 2025.
He has since played 96 times for the club, scoring nine goals with 12 assists, and was part of the side that won the Sao Paulo top-flight Campeonato Paulista in 2024 and 2026.
He is expected to miss Wednesday's Copa do Brasil quarter-final against Santos as he prepares for the move, according to Brazilian media reports.
Allan is set to replace fellow Brazilian Savinho, who is reportedly nearing a move from City to Tottenham.