However, Bournemouth's Eddie Howe, one of the early favourites to take over at St. Mary's, is said to have been cut from the Saints' shortlist.
It is claimed that both parties were uneasy about the move, which would have seen Howe move 30 miles up the south coast.
Howe has led the Cherries to three promotions
Garcia, meanwhile, has shot straight to the top of the list of bookies' favourites, having led Roma to Champions League qualification in each of his two full seasons at the Olympic Stadium.
Despite this, the Frenchman was sacked in January following a poor run of form and is currently a free agent.
Garcia has managed at the highest level
Previously, Garcia spent five years in charge of Lille, where he delivered a league and cup double in 2010-11.
the Telegraph reports that Frank de Boer remains in contention, having left Ajax in May, but former Chelsea and Tottenham Hotspur boss Andre Villas Boas was never considered a viable candidate.
Koeman leaves behind sizeable shoes to fill, having led the Saints to record Premier League finishes in each of his two seasons as boss.
Koeman left for Southampton on Tuesday