An agreement between the Dutch and Spanish clubs is understood to be close, according to the BBC.
Parrott, 24, began his career at Tottenham Hotspur before joining Alkmaar in July 2024, having spent the previous season on loan at fellow Dutch side Excelsior Rotterdam.
His form in the Netherlands has been central to Betis' interest. Parrott scored 10 goals in his first seven appearances of last season before a knee ligament injury sustained at the end of August ruled him out until October. He finished the Eredivisie campaign with 16 goals in 28 league games.
The striker has also become a key figure for his country. He scored both goals in a 2-0 win over Portugal last November, then added a hat-trick three days later, including a 96th-minute winner, in a 3-2 victory away to Hungary.
The treble made him the first Republic of Ireland player to score a hat-trick in an away game for the men's senior team, and the first Irish player to score a competitive treble since Robbie Keane in 2014, as Ireland secured their place in the World Cup play-offs.
Capped 37 times, Parrott is expected to remain a pivotal figure for Heimir Hallgrimsson's side in the upcoming Nations League, where the Republic of Ireland face Kosovo, Austria and Israel in Group B3.
A move to Betis would offer Parrott Champions League football, with the club having finished fifth in La Liga last season to secure qualification for Europe's premier club competition. That prospect appears to have given the Spanish side the edge over West Ham, who could offer only Championship football following their relegation from the Premier League.
AZ Alkmaar signed Parrott for a reported €4million from Tottenham in 2024, and the proposed €20million fee would represent a significant return for the Dutch club on that outlay. Parrott's contract at Alkmaar runs until the summer of 2029, giving the club a strong negotiating position despite their willingness to sell.