The Saints have endured a tricky start to life back at the top table of English football following their playoff success in the Championship last season.
The South Coast club have collected just one point from their opening five Premier League contests of the 2024-25 schedule, yet to enjoy a maiden win of the campaign.
Martin's troops came mightily close to that first maximum on Saturday afternoon, when a first-ever professional goal from Tyler Dibling put the hosts ahead at St Mary's Stadium against Ipswich Town.
However, the visiting Tractor Boys were not ready to lie down and accept defeat, with midfielder Sam Morsy netting a sensational 95th-minute equaliser.
Southampton to give Martin two more games?
According to Football Insider, Southampton have finalised their strategy concerning the future of Martin as the club's wait for a first win continues.
The report claims that the Saints owners are willing to give the 38-year-old a further two games to pick up a win before wielding the managerial axe.
It is understood that Southampton's hierarchy is growing tired of the team's expansive style, one which has cost them on numerous occasions so far this season.
High-ranking members of the Saints board want Martin to employ a more pragmatic style of play in order to grind out Premier League victories.
It is believed that a list of replacements is currently in the process of being drawn up, with Blackburn Rovers boss John Eustace one of the names linked to the role.
Tricky pair of games for the Saints
With Southampton's board set to give Martin until the October international break to turn the team's fortunes around, the Scotsman will be laser focused on the pair of matches before the pause.
First up for the Saints is a trip across the South Coast to Bournemouth on Saturday afternoon, with the Cherries winning just one of their first five contests of the season.
Should they fail to collect a win at the Vitality Stadium, it is likely that a daunting trip to Arsenal on October 5 could be the final game of Martin's reign at Southampton.